(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0) → ok(0)
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
isNat(ok(z0)) → ok(isNat(z0))
isNatKind(ok(z0)) → ok(isNatKind(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(U11(tt, z0, z1)) → c(U12'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U12(tt, z0)) → c1(U13'(isNat(z0)), ISNAT(z0))
ACTIVE(U13(tt)) → c2
ACTIVE(U21(tt, z0)) → c3(U22'(isNat(z0)), ISNAT(z0))
ACTIVE(U22(tt)) → c4
ACTIVE(U31(tt, z0)) → c5
ACTIVE(U41(tt, z0, z1)) → c6(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(and(tt, z0)) → c7
ACTIVE(isNat(0)) → c8
ACTIVE(isNat(plus(z0, z1))) → c9(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNat(s(z0))) → c10(U21'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(isNatKind(0)) → c11
ACTIVE(isNatKind(plus(z0, z1))) → c12(AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, 0)) → c14(U31'(and(isNat(z0), isNatKind(z0)), z0), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c15(U41'(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), AND(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), AND(isNat(z1), isNatKind(z1)), ISNAT(z1), ISNATKIND(z1), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(U11(z0, z1, z2)) → c16(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(tt) → c49
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c60
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
S tuples:
ACTIVE(U11(tt, z0, z1)) → c(U12'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U12(tt, z0)) → c1(U13'(isNat(z0)), ISNAT(z0))
ACTIVE(U13(tt)) → c2
ACTIVE(U21(tt, z0)) → c3(U22'(isNat(z0)), ISNAT(z0))
ACTIVE(U22(tt)) → c4
ACTIVE(U31(tt, z0)) → c5
ACTIVE(U41(tt, z0, z1)) → c6(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(and(tt, z0)) → c7
ACTIVE(isNat(0)) → c8
ACTIVE(isNat(plus(z0, z1))) → c9(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNat(s(z0))) → c10(U21'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(isNatKind(0)) → c11
ACTIVE(isNatKind(plus(z0, z1))) → c12(AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, 0)) → c14(U31'(and(isNat(z0), isNatKind(z0)), z0), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c15(U41'(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), AND(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), AND(isNat(z1), isNatKind(z1)), ISNAT(z1), ISNATKIND(z1), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(U11(z0, z1, z2)) → c16(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(tt) → c49
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c60
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:
active, U11, U12, U13, U21, U22, U31, U41, s, plus, and, proper, isNat, isNatKind, top
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 8 trailing nodes:
ACTIVE(isNatKind(0)) → c11
PROPER(tt) → c49
ACTIVE(U31(tt, z0)) → c5
PROPER(0) → c60
ACTIVE(U13(tt)) → c2
ACTIVE(U22(tt)) → c4
ACTIVE(and(tt, z0)) → c7
ACTIVE(isNat(0)) → c8
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
isNat(ok(z0)) → ok(isNat(z0))
isNatKind(ok(z0)) → ok(isNatKind(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(U11(tt, z0, z1)) → c(U12'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U12(tt, z0)) → c1(U13'(isNat(z0)), ISNAT(z0))
ACTIVE(U21(tt, z0)) → c3(U22'(isNat(z0)), ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c6(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c9(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNat(s(z0))) → c10(U21'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c12(AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, 0)) → c14(U31'(and(isNat(z0), isNatKind(z0)), z0), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c15(U41'(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), AND(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), AND(isNat(z1), isNatKind(z1)), ISNAT(z1), ISNATKIND(z1), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(U11(z0, z1, z2)) → c16(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
S tuples:
ACTIVE(U11(tt, z0, z1)) → c(U12'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U12(tt, z0)) → c1(U13'(isNat(z0)), ISNAT(z0))
ACTIVE(U21(tt, z0)) → c3(U22'(isNat(z0)), ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c6(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c9(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNat(s(z0))) → c10(U21'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c12(AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, 0)) → c14(U31'(and(isNat(z0), isNatKind(z0)), z0), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c15(U41'(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), AND(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), AND(isNat(z1), isNatKind(z1)), ISNAT(z1), ISNATKIND(z1), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(U11(z0, z1, z2)) → c16(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:
active, U11, U12, U13, U21, U22, U31, U41, s, plus, and, proper, isNat, isNatKind, top
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c, c1, c3, c6, c9, c10, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
isNat(ok(z0)) → ok(isNat(z0))
isNatKind(ok(z0)) → ok(isNatKind(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(U11(tt, z0, z1)) → c(U12'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U12(tt, z0)) → c1(U13'(isNat(z0)), ISNAT(z0))
ACTIVE(U21(tt, z0)) → c3(U22'(isNat(z0)), ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c6(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c9(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c12(AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, 0)) → c14(U31'(and(isNat(z0), isNatKind(z0)), z0), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(U11(z0, z1, z2)) → c16(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c15(ISNAT(z1), ISNATKIND(z1), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
S tuples:
ACTIVE(U11(tt, z0, z1)) → c(U12'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U12(tt, z0)) → c1(U13'(isNat(z0)), ISNAT(z0))
ACTIVE(U21(tt, z0)) → c3(U22'(isNat(z0)), ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c6(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c9(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c12(AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, 0)) → c14(U31'(and(isNat(z0), isNatKind(z0)), z0), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(U11(z0, z1, z2)) → c16(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c15(ISNAT(z1), ISNATKIND(z1), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
K tuples:none
Defined Rule Symbols:
active, U11, U12, U13, U21, U22, U31, U41, s, plus, and, proper, isNat, isNatKind, top
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c, c1, c3, c6, c9, c12, c13, c14, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c15
(7) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
isNat(ok(z0)) → ok(isNat(z0))
isNatKind(ok(z0)) → ok(isNatKind(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U11(z0, z1, z2)) → c16(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U11(z0, z1, z2)) → c16(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
K tuples:none
Defined Rule Symbols:
active, U11, U12, U13, U21, U22, U31, U41, s, plus, and, proper, isNat, isNatKind, top
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2
(9) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U11(z0, z1, z2)) → c16(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U11(z0, z1, z2)) → c16(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2
(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
U11(
z0,
z1,
z2)) →
c16(
U11'(
active(
z0),
z1,
z2),
ACTIVE(
z0)) by
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c16(U11'(mark(U12(isNat(z0), z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c16(U11'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c16(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c16(U11'(mark(U21(isNatKind(z0), z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(isNatKind(0)))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c16(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c16(U11'(mark(isNatKind(z0)), x1, x2), ACTIVE(isNatKind(s(z0))))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c16(U11'(mark(U12(isNat(z0), z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c16(U11'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c16(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c16(U11'(mark(U21(isNatKind(z0), z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(isNatKind(0)))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c16(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c16(U11'(mark(isNatKind(z0)), x1, x2), ACTIVE(isNatKind(s(z0))))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c16(U11'(mark(U12(isNat(z0), z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c16(U11'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c16(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c16(U11'(mark(U21(isNatKind(z0), z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(isNatKind(0)))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c16(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c16(U11'(mark(isNatKind(z0)), x1, x2), ACTIVE(isNatKind(s(z0))))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16
(13) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c16(U11'(mark(U12(isNat(z0), z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c16(U11'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c16(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c16(U11'(mark(U21(isNatKind(z0), z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c16(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c16(U11'(mark(isNatKind(z0)), x1, x2), ACTIVE(isNatKind(s(z0))))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c16(U11'(mark(U12(isNat(z0), z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c16(U11'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c16(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c16(U11'(mark(U21(isNatKind(z0), z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c16(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c16(U11'(mark(isNatKind(z0)), x1, x2), ACTIVE(isNatKind(s(z0))))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16
(15) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c16(U11'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U12(z0, z1)) → c17(U12'(active(z0), z1), ACTIVE(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c16(U11'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c
(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
U12(
z0,
z1)) →
c17(
U12'(
active(
z0),
z1),
ACTIVE(
z0)) by
ACTIVE(U12(U11(tt, z0, z1), x1)) → c17(U12'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c17(U12'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c17(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c17(U12'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1), ACTIVE(isNatKind(0)))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c17(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c17(U12'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
(18) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c16(U11'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c17(U12'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c17(U12'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c17(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c17(U12'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1), ACTIVE(isNatKind(0)))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c17(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c17(U12'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c16(U11'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c17(U12'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c17(U12'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c17(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c17(U12'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1), ACTIVE(isNatKind(0)))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c17(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c17(U12'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17
(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(20) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c16(U11'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c17(U12'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c17(U12'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c17(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c17(U12'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c17(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c17(U12'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c16(U11'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c17(U12'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c17(U12'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c17(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c17(U12'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c17(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c17(U12'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17
(21) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(22) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U13(z0)) → c18(U13'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1
(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
U13(
z0)) →
c18(
U13'(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(U13(U11(tt, z0, z1))) → c18(U13'(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c18(U13'(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)), ACTIVE(U13(tt)))
ACTIVE(U13(U21(tt, z0))) → c18(U13'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c18(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c18(U13'(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)), ACTIVE(isNatKind(0)))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c18(U13'(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c18(U13'(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
(24) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U11(tt, z0, z1))) → c18(U13'(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c18(U13'(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)), ACTIVE(U13(tt)))
ACTIVE(U13(U21(tt, z0))) → c18(U13'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c18(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c18(U13'(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)), ACTIVE(isNatKind(0)))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c18(U13'(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c18(U13'(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U11(tt, z0, z1))) → c18(U13'(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c18(U13'(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)), ACTIVE(U13(tt)))
ACTIVE(U13(U21(tt, z0))) → c18(U13'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c18(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c18(U13'(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)), ACTIVE(isNatKind(0)))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c18(U13'(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c18(U13'(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18
(25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing tuple parts
(26) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U11(tt, z0, z1))) → c18(U13'(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c18(U13'(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c18(U13'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c18(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c18(U13'(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c18(U13'(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c18(U13'(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U11(tt, z0, z1))) → c18(U13'(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c18(U13'(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c18(U13'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c18(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c18(U13'(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c18(U13'(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c18(U13'(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18
(27) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(28) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U21(tt, z0))) → c18(U13'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U21(z0, z1)) → c19(U21'(active(z0), z1), ACTIVE(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U21(tt, z0))) → c18(U13'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3
(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
U21(
z0,
z1)) →
c19(
U21'(
active(
z0),
z1),
ACTIVE(
z0)) by
ACTIVE(U21(U11(tt, z0, z1), x1)) → c19(U21'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c19(U21'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U21(U21(tt, z0), x1)) → c19(U21'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c19(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c19(U21'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1), ACTIVE(isNatKind(0)))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c19(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c19(U21'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
(30) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U21(tt, z0))) → c18(U13'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c19(U21'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c19(U21'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U21(U21(tt, z0), x1)) → c19(U21'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c19(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c19(U21'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1), ACTIVE(isNatKind(0)))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c19(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c19(U21'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U21(tt, z0))) → c18(U13'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c19(U21'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c19(U21'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U21(U21(tt, z0), x1)) → c19(U21'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c19(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c19(U21'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1), ACTIVE(isNatKind(0)))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c19(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c19(U21'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19
(31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing tuple parts
(32) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U21(tt, z0))) → c18(U13'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c19(U21'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c19(U21'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c19(U21'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c19(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c19(U21'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c19(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c19(U21'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c16(U11'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U21(tt, z0), x1)) → c17(U12'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U21(tt, z0))) → c18(U13'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c19(U21'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c19(U21'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c19(U21'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c19(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c19(U21'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c19(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c19(U21'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19
(33) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(34) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U22(z0)) → c20(U22'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4
(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
U22(
z0)) →
c20(
U22'(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(U22(U11(tt, z0, z1))) → c20(U22'(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c20(U22'(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)), ACTIVE(U13(tt)))
ACTIVE(U22(U21(tt, z0))) → c20(U22'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c20(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c20(U22'(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)), ACTIVE(isNatKind(0)))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c20(U22'(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c20(U22'(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
(36) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U11(tt, z0, z1))) → c20(U22'(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c20(U22'(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)), ACTIVE(U13(tt)))
ACTIVE(U22(U21(tt, z0))) → c20(U22'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c20(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c20(U22'(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)), ACTIVE(isNatKind(0)))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c20(U22'(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c20(U22'(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U11(tt, z0, z1))) → c20(U22'(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c20(U22'(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)), ACTIVE(U13(tt)))
ACTIVE(U22(U21(tt, z0))) → c20(U22'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c20(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c20(U22'(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)), ACTIVE(isNatKind(0)))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c20(U22'(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c20(U22'(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20
(37) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 8 trailing tuple parts
(38) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U11(tt, z0, z1))) → c20(U22'(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c20(U22'(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c20(U22'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c20(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c20(U22'(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c20(U22'(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c20(U22'(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U11(tt, z0, z1))) → c20(U22'(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c20(U22'(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c20(U22'(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c20(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c20(U22'(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c20(U22'(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c20(U22'(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20
(39) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(40) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U31(z0, z1)) → c21(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5
(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
U31(
z0,
z1)) →
c21(
U31'(
active(
z0),
z1),
ACTIVE(
z0)) by
ACTIVE(U31(U11(tt, z0, z1), x1)) → c21(U31'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c21(U31'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U31(U21(tt, z0), x1)) → c21(U31'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c21(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c21(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c21(U31'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1), ACTIVE(isNatKind(0)))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c21(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c21(U31'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
(42) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c21(U31'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c21(U31'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U31(U21(tt, z0), x1)) → c21(U31'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c21(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c21(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c21(U31'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1), ACTIVE(isNatKind(0)))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c21(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c21(U31'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c21(U31'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c21(U31'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(U31(U21(tt, z0), x1)) → c21(U31'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c21(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c21(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c21(U31'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1), ACTIVE(isNatKind(0)))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c21(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c21(U31'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21
(43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 10 trailing tuple parts
(44) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c21(U31'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c21(U31'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c21(U31'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c21(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c21(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c21(U31'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c21(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c21(U31'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c21(U31'(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c21(U31'(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c21(U31'(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c21(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c21(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c21(U31'(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c21(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c21(U31'(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21
(45) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(46) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c21(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(U41(z0, z1, z2)) → c22(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c21(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6
(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
U41(
z0,
z1,
z2)) →
c22(
U41'(
active(
z0),
z1,
z2),
ACTIVE(
z0)) by
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c22(U41'(mark(U12(isNat(z0), z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c22(U41'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c22(U41'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c22(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c22(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c22(U41'(mark(U21(isNatKind(z0), z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2), ACTIVE(isNatKind(0)))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c22(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c22(U41'(mark(isNatKind(z0)), x1, x2), ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
(48) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c21(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c22(U41'(mark(U12(isNat(z0), z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c22(U41'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c22(U41'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c22(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c22(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c22(U41'(mark(U21(isNatKind(z0), z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2), ACTIVE(isNatKind(0)))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c22(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c22(U41'(mark(isNatKind(z0)), x1, x2), ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c21(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c22(U41'(mark(U12(isNat(z0), z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c22(U41'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2), ACTIVE(U13(tt)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c22(U41'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2), ACTIVE(U22(tt)))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c22(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c22(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c22(U41'(mark(U21(isNatKind(z0), z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2), ACTIVE(isNatKind(0)))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c22(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c22(U41'(mark(isNatKind(z0)), x1, x2), ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22
(49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing tuple parts
(50) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c21(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c22(U41'(mark(U12(isNat(z0), z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c22(U41'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c22(U41'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c22(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c22(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c22(U41'(mark(U21(isNatKind(z0), z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c22(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c22(U41'(mark(isNatKind(z0)), x1, x2), ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c16(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c17(U12'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(U41(tt, z0, z1))) → c18(U13'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c19(U21'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(U41(tt, z0, z1))) → c20(U22'(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c21(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c22(U41'(mark(U12(isNat(z0), z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c22(U41'(mark(U13(isNat(z0))), x1, x2), ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c22(U41'(mark(U22(isNat(z0))), x1, x2), ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c22(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c22(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c22(U41'(mark(U21(isNatKind(z0), z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c22(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c22(U41'(mark(isNatKind(z0)), x1, x2), ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22
(51) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(52) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(s(z0)) → c23(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7
(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
s(
z0)) →
c23(
S(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(s(U11(tt, z0, z1))) → c23(S(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c23(S(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)), ACTIVE(U13(tt)))
ACTIVE(s(U21(tt, z0))) → c23(S(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c23(S(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c23(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c23(S(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)), ACTIVE(isNatKind(0)))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c23(S(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c23(S(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
(54) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(U11(tt, z0, z1))) → c23(S(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c23(S(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)), ACTIVE(U13(tt)))
ACTIVE(s(U21(tt, z0))) → c23(S(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c23(S(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c23(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c23(S(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)), ACTIVE(isNatKind(0)))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c23(S(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c23(S(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(U11(tt, z0, z1))) → c23(S(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c23(S(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)), ACTIVE(U13(tt)))
ACTIVE(s(U21(tt, z0))) → c23(S(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)), ACTIVE(U22(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c23(S(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c23(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c23(S(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)), ACTIVE(isNatKind(0)))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c23(S(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c23(S(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23
(55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing tuple parts
(56) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(U11(tt, z0, z1))) → c23(S(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c23(S(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c23(S(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c23(S(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c23(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c23(S(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c23(S(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c23(S(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(U11(tt, z0, z1))) → c23(S(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c23(S(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c23(S(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c23(S(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c23(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c23(S(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c23(S(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c23(S(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23
(57) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(58) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c26(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8
(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
and(
z0,
z1)) →
c26(
AND(
active(
z0),
z1),
ACTIVE(
z0)) by
ACTIVE(and(U11(tt, z0, z1), x1)) → c26(AND(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c26(AND(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(and(U21(tt, z0), x1)) → c26(AND(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c26(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c26(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c26(AND(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1), ACTIVE(isNatKind(0)))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c26(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c26(AND(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
(60) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(U11(tt, z0, z1), x1)) → c26(AND(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c26(AND(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(and(U21(tt, z0), x1)) → c26(AND(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c26(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c26(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c26(AND(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1), ACTIVE(isNatKind(0)))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c26(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c26(AND(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(U11(tt, z0, z1), x1)) → c26(AND(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c26(AND(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1), ACTIVE(U13(tt)))
ACTIVE(and(U21(tt, z0), x1)) → c26(AND(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1), ACTIVE(U22(tt)))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c26(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c26(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c26(AND(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1), ACTIVE(isNatKind(0)))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c26(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c26(AND(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26
(61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 14 trailing tuple parts
(62) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(U11(tt, z0, z1), x1)) → c26(AND(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c26(AND(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c26(AND(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c26(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c26(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c26(AND(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c26(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c26(AND(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(U11(tt, z0, z1), x1)) → c26(AND(mark(U12(isNat(z0), z1)), x1), ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c26(AND(mark(U13(isNat(z0))), x1), ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c26(AND(mark(U22(isNat(z0))), x1), ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c26(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c26(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c26(AND(mark(U21(isNatKind(z0), z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c26(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1), ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c26(AND(mark(isNatKind(z0)), x1), ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26
(63) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(64) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U11(z0, z1, z2)) → c48(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9
(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
U11(
z0,
z1,
z2)) →
c48(
U11'(
proper(
z0),
proper(
z1),
proper(
z2)),
PROPER(
z0),
PROPER(
z1),
PROPER(
z2)) by
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
(66) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48
(67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing tuple parts
(68) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U12(z0, z1)) → c50(U12'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48
(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
U12(
z0,
z1)) →
c50(
U12'(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
(70) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50
(71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(72) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c51, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50
(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
isNat(
z0)) →
c51(
ISNAT(
proper(
z0)),
PROPER(
z0)) by
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)), PROPER(tt))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)), PROPER(0))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
(74) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)), PROPER(tt))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)), PROPER(0))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)), PROPER(tt))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)), PROPER(0))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51
(75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(76) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U13(z0)) → c52(U13'(proper(z0)), PROPER(z0))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c52, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51
(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
U13(
z0)) →
c52(
U13'(
proper(
z0)),
PROPER(
z0)) by
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(tt)) → c52(U13'(ok(tt)), PROPER(tt))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(0)) → c52(U13'(ok(0)), PROPER(0))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
(78) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(tt)) → c52(U13'(ok(tt)), PROPER(tt))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(0)) → c52(U13'(ok(0)), PROPER(0))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(tt)) → c52(U13'(ok(tt)), PROPER(tt))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(0)) → c52(U13'(ok(0)), PROPER(0))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52
(79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(80) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U21(z0, z1)) → c53(U21'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c53, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52
(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
U21(
z0,
z1)) →
c53(
U21'(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
(82) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53
(83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(84) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U22(z0)) → c54(U22'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c54, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53
(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
U22(
z0)) →
c54(
U22'(
proper(
z0)),
PROPER(
z0)) by
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(tt)) → c54(U22'(ok(tt)), PROPER(tt))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(0)) → c54(U22'(ok(0)), PROPER(0))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
(86) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(tt)) → c54(U22'(ok(tt)), PROPER(tt))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(0)) → c54(U22'(ok(0)), PROPER(0))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(tt)) → c54(U22'(ok(tt)), PROPER(tt))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(0)) → c54(U22'(ok(0)), PROPER(0))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54
(87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(88) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U31(z0, z1)) → c55(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c55, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54
(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
U31(
z0,
z1)) →
c55(
U31'(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
(90) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55
(91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(92) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(U41(z0, z1, z2)) → c56(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c56, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55
(93) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
U41(
z0,
z1,
z2)) →
c56(
U41'(
proper(
z0),
proper(
z1),
proper(
z2)),
PROPER(
z0),
PROPER(
z1),
PROPER(
z2)) by
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
(94) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55, c56
(95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing tuple parts
(96) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(s(z0)) → c57(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c57, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55, c56, c56
(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
s(
z0)) →
c57(
S(
proper(
z0)),
PROPER(
z0)) by
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(tt)) → c57(S(ok(tt)), PROPER(tt))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(0)) → c57(S(ok(0)), PROPER(0))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
(98) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(tt)) → c57(S(ok(tt)), PROPER(tt))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(0)) → c57(S(ok(0)), PROPER(0))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(tt)) → c57(S(ok(tt)), PROPER(tt))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(0)) → c57(S(ok(0)), PROPER(0))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55, c56, c56, c57
(99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(100) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(plus(z0, z1)) → c58(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c58, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55, c56, c56, c57, c57
(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
plus(
z0,
z1)) →
c58(
PLUS(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
(102) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55, c56, c56, c57, c57, c58
(103) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(104) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(and(z0, z1)) → c59(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c59, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55, c56, c56, c57, c57, c58, c58
(105) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
and(
z0,
z1)) →
c59(
AND(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
(106) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55, c56, c56, c57, c57, c58, c58, c59
(107) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(108) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
PROPER(isNatKind(z0)) → c61(ISNATKIND(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, PROPER, ISNAT, ISNATKIND, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c61, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55, c56, c56, c57, c57, c58, c58, c59, c59
(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
isNatKind(
z0)) →
c61(
ISNATKIND(
proper(
z0)),
PROPER(
z0)) by
PROPER(isNatKind(U11(z0, z1, z2))) → c61(ISNATKIND(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNatKind(tt)) → c61(ISNATKIND(ok(tt)), PROPER(tt))
PROPER(isNatKind(U12(z0, z1))) → c61(ISNATKIND(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNatKind(isNat(z0))) → c61(ISNATKIND(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNatKind(U13(z0))) → c61(ISNATKIND(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNatKind(U21(z0, z1))) → c61(ISNATKIND(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNatKind(U22(z0))) → c61(ISNATKIND(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNatKind(U31(z0, z1))) → c61(ISNATKIND(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNatKind(U41(z0, z1, z2))) → c61(ISNATKIND(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNatKind(s(z0))) → c61(ISNATKIND(s(proper(z0))), PROPER(s(z0)))
PROPER(isNatKind(plus(z0, z1))) → c61(ISNATKIND(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNatKind(and(z0, z1))) → c61(ISNATKIND(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNatKind(0)) → c61(ISNATKIND(ok(0)), PROPER(0))
PROPER(isNatKind(isNatKind(z0))) → c61(ISNATKIND(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
(110) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNatKind(U11(z0, z1, z2))) → c61(ISNATKIND(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNatKind(tt)) → c61(ISNATKIND(ok(tt)), PROPER(tt))
PROPER(isNatKind(U12(z0, z1))) → c61(ISNATKIND(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNatKind(isNat(z0))) → c61(ISNATKIND(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNatKind(U13(z0))) → c61(ISNATKIND(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNatKind(U21(z0, z1))) → c61(ISNATKIND(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNatKind(U22(z0))) → c61(ISNATKIND(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNatKind(U31(z0, z1))) → c61(ISNATKIND(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNatKind(U41(z0, z1, z2))) → c61(ISNATKIND(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNatKind(s(z0))) → c61(ISNATKIND(s(proper(z0))), PROPER(s(z0)))
PROPER(isNatKind(plus(z0, z1))) → c61(ISNATKIND(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNatKind(and(z0, z1))) → c61(ISNATKIND(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNatKind(0)) → c61(ISNATKIND(ok(0)), PROPER(0))
PROPER(isNatKind(isNatKind(z0))) → c61(ISNATKIND(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNatKind(U11(z0, z1, z2))) → c61(ISNATKIND(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNatKind(tt)) → c61(ISNATKIND(ok(tt)), PROPER(tt))
PROPER(isNatKind(U12(z0, z1))) → c61(ISNATKIND(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNatKind(isNat(z0))) → c61(ISNATKIND(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNatKind(U13(z0))) → c61(ISNATKIND(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNatKind(U21(z0, z1))) → c61(ISNATKIND(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNatKind(U22(z0))) → c61(ISNATKIND(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNatKind(U31(z0, z1))) → c61(ISNATKIND(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNatKind(U41(z0, z1, z2))) → c61(ISNATKIND(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNatKind(s(z0))) → c61(ISNATKIND(s(proper(z0))), PROPER(s(z0)))
PROPER(isNatKind(plus(z0, z1))) → c61(ISNATKIND(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNatKind(and(z0, z1))) → c61(ISNATKIND(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNatKind(0)) → c61(ISNATKIND(ok(0)), PROPER(0))
PROPER(isNatKind(isNatKind(z0))) → c61(ISNATKIND(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND, TOP, PROPER
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55, c56, c56, c57, c57, c58, c58, c59, c59, c61
(111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(112) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNatKind(U11(z0, z1, z2))) → c61(ISNATKIND(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNatKind(U12(z0, z1))) → c61(ISNATKIND(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNatKind(isNat(z0))) → c61(ISNATKIND(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNatKind(U13(z0))) → c61(ISNATKIND(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNatKind(U21(z0, z1))) → c61(ISNATKIND(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNatKind(U22(z0))) → c61(ISNATKIND(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNatKind(U31(z0, z1))) → c61(ISNATKIND(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNatKind(U41(z0, z1, z2))) → c61(ISNATKIND(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNatKind(s(z0))) → c61(ISNATKIND(s(proper(z0))), PROPER(s(z0)))
PROPER(isNatKind(plus(z0, z1))) → c61(ISNATKIND(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNatKind(and(z0, z1))) → c61(ISNATKIND(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNatKind(isNatKind(z0))) → c61(ISNATKIND(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNatKind(tt)) → c61(ISNATKIND(ok(tt)))
PROPER(isNatKind(0)) → c61(ISNATKIND(ok(0)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(mark(z0)) → c64(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNatKind(U11(z0, z1, z2))) → c61(ISNATKIND(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNatKind(U12(z0, z1))) → c61(ISNATKIND(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNatKind(isNat(z0))) → c61(ISNATKIND(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNatKind(U13(z0))) → c61(ISNATKIND(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNatKind(U21(z0, z1))) → c61(ISNATKIND(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNatKind(U22(z0))) → c61(ISNATKIND(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNatKind(U31(z0, z1))) → c61(ISNATKIND(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNatKind(U41(z0, z1, z2))) → c61(ISNATKIND(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNatKind(s(z0))) → c61(ISNATKIND(s(proper(z0))), PROPER(s(z0)))
PROPER(isNatKind(plus(z0, z1))) → c61(ISNATKIND(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNatKind(and(z0, z1))) → c61(ISNATKIND(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNatKind(isNatKind(z0))) → c61(ISNATKIND(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNatKind(tt)) → c61(ISNATKIND(ok(tt)))
PROPER(isNatKind(0)) → c61(ISNATKIND(ok(0)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND, TOP, PROPER
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63, c64, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55, c56, c56, c57, c57, c58, c58, c59, c59, c61, c61
(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
TOP(
mark(
z0)) →
c64(
TOP(
proper(
z0)),
PROPER(
z0)) by
TOP(mark(U11(z0, z1, z2))) → c64(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(tt)) → c64(TOP(ok(tt)), PROPER(tt))
TOP(mark(U12(z0, z1))) → c64(TOP(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
TOP(mark(isNat(z0))) → c64(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U13(z0))) → c64(TOP(U13(proper(z0))), PROPER(U13(z0)))
TOP(mark(U21(z0, z1))) → c64(TOP(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
TOP(mark(U22(z0))) → c64(TOP(U22(proper(z0))), PROPER(U22(z0)))
TOP(mark(U31(z0, z1))) → c64(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c64(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(s(z0))) → c64(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c64(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c64(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(0)) → c64(TOP(ok(0)), PROPER(0))
TOP(mark(isNatKind(z0))) → c64(TOP(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
(114) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNatKind(U11(z0, z1, z2))) → c61(ISNATKIND(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNatKind(U12(z0, z1))) → c61(ISNATKIND(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNatKind(isNat(z0))) → c61(ISNATKIND(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNatKind(U13(z0))) → c61(ISNATKIND(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNatKind(U21(z0, z1))) → c61(ISNATKIND(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNatKind(U22(z0))) → c61(ISNATKIND(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNatKind(U31(z0, z1))) → c61(ISNATKIND(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNatKind(U41(z0, z1, z2))) → c61(ISNATKIND(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNatKind(s(z0))) → c61(ISNATKIND(s(proper(z0))), PROPER(s(z0)))
PROPER(isNatKind(plus(z0, z1))) → c61(ISNATKIND(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNatKind(and(z0, z1))) → c61(ISNATKIND(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNatKind(isNatKind(z0))) → c61(ISNATKIND(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNatKind(tt)) → c61(ISNATKIND(ok(tt)))
PROPER(isNatKind(0)) → c61(ISNATKIND(ok(0)))
TOP(mark(U11(z0, z1, z2))) → c64(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(tt)) → c64(TOP(ok(tt)), PROPER(tt))
TOP(mark(U12(z0, z1))) → c64(TOP(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
TOP(mark(isNat(z0))) → c64(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U13(z0))) → c64(TOP(U13(proper(z0))), PROPER(U13(z0)))
TOP(mark(U21(z0, z1))) → c64(TOP(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
TOP(mark(U22(z0))) → c64(TOP(U22(proper(z0))), PROPER(U22(z0)))
TOP(mark(U31(z0, z1))) → c64(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c64(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(s(z0))) → c64(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c64(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c64(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(0)) → c64(TOP(ok(0)), PROPER(0))
TOP(mark(isNatKind(z0))) → c64(TOP(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNatKind(U11(z0, z1, z2))) → c61(ISNATKIND(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNatKind(U12(z0, z1))) → c61(ISNATKIND(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNatKind(isNat(z0))) → c61(ISNATKIND(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNatKind(U13(z0))) → c61(ISNATKIND(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNatKind(U21(z0, z1))) → c61(ISNATKIND(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNatKind(U22(z0))) → c61(ISNATKIND(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNatKind(U31(z0, z1))) → c61(ISNATKIND(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNatKind(U41(z0, z1, z2))) → c61(ISNATKIND(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNatKind(s(z0))) → c61(ISNATKIND(s(proper(z0))), PROPER(s(z0)))
PROPER(isNatKind(plus(z0, z1))) → c61(ISNATKIND(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNatKind(and(z0, z1))) → c61(ISNATKIND(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNatKind(isNatKind(z0))) → c61(ISNATKIND(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNatKind(tt)) → c61(ISNATKIND(ok(tt)))
PROPER(isNatKind(0)) → c61(ISNATKIND(ok(0)))
TOP(mark(U11(z0, z1, z2))) → c64(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(tt)) → c64(TOP(ok(tt)), PROPER(tt))
TOP(mark(U12(z0, z1))) → c64(TOP(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
TOP(mark(isNat(z0))) → c64(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U13(z0))) → c64(TOP(U13(proper(z0))), PROPER(U13(z0)))
TOP(mark(U21(z0, z1))) → c64(TOP(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
TOP(mark(U22(z0))) → c64(TOP(U22(proper(z0))), PROPER(U22(z0)))
TOP(mark(U31(z0, z1))) → c64(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c64(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(s(z0))) → c64(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c64(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c64(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(0)) → c64(TOP(ok(0)), PROPER(0))
TOP(mark(isNatKind(z0))) → c64(TOP(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND, TOP, PROPER
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55, c56, c56, c57, c57, c58, c58, c59, c59, c61, c61, c64
(115) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(116) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNatKind(U11(z0, z1, z2))) → c61(ISNATKIND(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNatKind(U12(z0, z1))) → c61(ISNATKIND(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNatKind(isNat(z0))) → c61(ISNATKIND(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNatKind(U13(z0))) → c61(ISNATKIND(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNatKind(U21(z0, z1))) → c61(ISNATKIND(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNatKind(U22(z0))) → c61(ISNATKIND(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNatKind(U31(z0, z1))) → c61(ISNATKIND(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNatKind(U41(z0, z1, z2))) → c61(ISNATKIND(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNatKind(s(z0))) → c61(ISNATKIND(s(proper(z0))), PROPER(s(z0)))
PROPER(isNatKind(plus(z0, z1))) → c61(ISNATKIND(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNatKind(and(z0, z1))) → c61(ISNATKIND(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNatKind(isNatKind(z0))) → c61(ISNATKIND(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNatKind(tt)) → c61(ISNATKIND(ok(tt)))
PROPER(isNatKind(0)) → c61(ISNATKIND(ok(0)))
TOP(mark(U11(z0, z1, z2))) → c64(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(U12(z0, z1))) → c64(TOP(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
TOP(mark(isNat(z0))) → c64(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U13(z0))) → c64(TOP(U13(proper(z0))), PROPER(U13(z0)))
TOP(mark(U21(z0, z1))) → c64(TOP(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
TOP(mark(U22(z0))) → c64(TOP(U22(proper(z0))), PROPER(U22(z0)))
TOP(mark(U31(z0, z1))) → c64(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c64(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(s(z0))) → c64(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c64(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c64(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNatKind(z0))) → c64(TOP(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
TOP(mark(tt)) → c64(TOP(ok(tt)))
TOP(mark(0)) → c64(TOP(ok(0)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
TOP(ok(z0)) → c65(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNatKind(U11(z0, z1, z2))) → c61(ISNATKIND(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNatKind(U12(z0, z1))) → c61(ISNATKIND(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNatKind(isNat(z0))) → c61(ISNATKIND(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNatKind(U13(z0))) → c61(ISNATKIND(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNatKind(U21(z0, z1))) → c61(ISNATKIND(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNatKind(U22(z0))) → c61(ISNATKIND(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNatKind(U31(z0, z1))) → c61(ISNATKIND(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNatKind(U41(z0, z1, z2))) → c61(ISNATKIND(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNatKind(s(z0))) → c61(ISNATKIND(s(proper(z0))), PROPER(s(z0)))
PROPER(isNatKind(plus(z0, z1))) → c61(ISNATKIND(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNatKind(and(z0, z1))) → c61(ISNATKIND(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNatKind(isNatKind(z0))) → c61(ISNATKIND(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNatKind(tt)) → c61(ISNATKIND(ok(tt)))
PROPER(isNatKind(0)) → c61(ISNATKIND(ok(0)))
TOP(mark(U11(z0, z1, z2))) → c64(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(U12(z0, z1))) → c64(TOP(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
TOP(mark(isNat(z0))) → c64(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U13(z0))) → c64(TOP(U13(proper(z0))), PROPER(U13(z0)))
TOP(mark(U21(z0, z1))) → c64(TOP(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
TOP(mark(U22(z0))) → c64(TOP(U22(proper(z0))), PROPER(U22(z0)))
TOP(mark(U31(z0, z1))) → c64(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c64(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(s(z0))) → c64(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c64(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c64(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNatKind(z0))) → c64(TOP(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
TOP(mark(tt)) → c64(TOP(ok(tt)))
TOP(mark(0)) → c64(TOP(ok(0)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND, TOP, PROPER
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63, c65, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55, c56, c56, c57, c57, c58, c58, c59, c59, c61, c61, c64, c64
(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
TOP(
ok(
z0)) →
c65(
TOP(
active(
z0)),
ACTIVE(
z0)) by
TOP(ok(U11(tt, z0, z1))) → c65(TOP(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
TOP(ok(U12(tt, z0))) → c65(TOP(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
TOP(ok(U13(tt))) → c65(TOP(mark(tt)), ACTIVE(U13(tt)))
TOP(ok(U21(tt, z0))) → c65(TOP(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
TOP(ok(U22(tt))) → c65(TOP(mark(tt)), ACTIVE(U22(tt)))
TOP(ok(U31(tt, z0))) → c65(TOP(mark(z0)), ACTIVE(U31(tt, z0)))
TOP(ok(U41(tt, z0, z1))) → c65(TOP(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c65(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(0))) → c65(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(ok(isNat(plus(z0, z1)))) → c65(TOP(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c65(TOP(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
TOP(ok(isNatKind(0))) → c65(TOP(mark(tt)), ACTIVE(isNatKind(0)))
TOP(ok(isNatKind(plus(z0, z1)))) → c65(TOP(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
TOP(ok(isNatKind(s(z0)))) → c65(TOP(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
TOP(ok(plus(z0, 0))) → c65(TOP(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c65(TOP(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1, z2))) → c65(TOP(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
TOP(ok(U12(z0, z1))) → c65(TOP(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
TOP(ok(U13(z0))) → c65(TOP(U13(active(z0))), ACTIVE(U13(z0)))
TOP(ok(U21(z0, z1))) → c65(TOP(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
TOP(ok(U22(z0))) → c65(TOP(U22(active(z0))), ACTIVE(U22(z0)))
TOP(ok(U31(z0, z1))) → c65(TOP(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
TOP(ok(U41(z0, z1, z2))) → c65(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(s(z0))) → c65(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c65(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c65(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(and(z0, z1))) → c65(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
(118) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNatKind(U11(z0, z1, z2))) → c61(ISNATKIND(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNatKind(U12(z0, z1))) → c61(ISNATKIND(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNatKind(isNat(z0))) → c61(ISNATKIND(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNatKind(U13(z0))) → c61(ISNATKIND(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNatKind(U21(z0, z1))) → c61(ISNATKIND(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNatKind(U22(z0))) → c61(ISNATKIND(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNatKind(U31(z0, z1))) → c61(ISNATKIND(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNatKind(U41(z0, z1, z2))) → c61(ISNATKIND(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNatKind(s(z0))) → c61(ISNATKIND(s(proper(z0))), PROPER(s(z0)))
PROPER(isNatKind(plus(z0, z1))) → c61(ISNATKIND(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNatKind(and(z0, z1))) → c61(ISNATKIND(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNatKind(isNatKind(z0))) → c61(ISNATKIND(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNatKind(tt)) → c61(ISNATKIND(ok(tt)))
PROPER(isNatKind(0)) → c61(ISNATKIND(ok(0)))
TOP(mark(U11(z0, z1, z2))) → c64(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(U12(z0, z1))) → c64(TOP(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
TOP(mark(isNat(z0))) → c64(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U13(z0))) → c64(TOP(U13(proper(z0))), PROPER(U13(z0)))
TOP(mark(U21(z0, z1))) → c64(TOP(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
TOP(mark(U22(z0))) → c64(TOP(U22(proper(z0))), PROPER(U22(z0)))
TOP(mark(U31(z0, z1))) → c64(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c64(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(s(z0))) → c64(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c64(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c64(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNatKind(z0))) → c64(TOP(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
TOP(mark(tt)) → c64(TOP(ok(tt)))
TOP(mark(0)) → c64(TOP(ok(0)))
TOP(ok(U11(tt, z0, z1))) → c65(TOP(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
TOP(ok(U12(tt, z0))) → c65(TOP(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
TOP(ok(U13(tt))) → c65(TOP(mark(tt)), ACTIVE(U13(tt)))
TOP(ok(U21(tt, z0))) → c65(TOP(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
TOP(ok(U22(tt))) → c65(TOP(mark(tt)), ACTIVE(U22(tt)))
TOP(ok(U31(tt, z0))) → c65(TOP(mark(z0)), ACTIVE(U31(tt, z0)))
TOP(ok(U41(tt, z0, z1))) → c65(TOP(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c65(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(0))) → c65(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(ok(isNat(plus(z0, z1)))) → c65(TOP(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c65(TOP(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
TOP(ok(isNatKind(0))) → c65(TOP(mark(tt)), ACTIVE(isNatKind(0)))
TOP(ok(isNatKind(plus(z0, z1)))) → c65(TOP(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
TOP(ok(isNatKind(s(z0)))) → c65(TOP(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
TOP(ok(plus(z0, 0))) → c65(TOP(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c65(TOP(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1, z2))) → c65(TOP(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
TOP(ok(U12(z0, z1))) → c65(TOP(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
TOP(ok(U13(z0))) → c65(TOP(U13(active(z0))), ACTIVE(U13(z0)))
TOP(ok(U21(z0, z1))) → c65(TOP(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
TOP(ok(U22(z0))) → c65(TOP(U22(active(z0))), ACTIVE(U22(z0)))
TOP(ok(U31(z0, z1))) → c65(TOP(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
TOP(ok(U41(z0, z1, z2))) → c65(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(s(z0))) → c65(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c65(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c65(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(and(z0, z1))) → c65(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNatKind(U11(z0, z1, z2))) → c61(ISNATKIND(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNatKind(U12(z0, z1))) → c61(ISNATKIND(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNatKind(isNat(z0))) → c61(ISNATKIND(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNatKind(U13(z0))) → c61(ISNATKIND(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNatKind(U21(z0, z1))) → c61(ISNATKIND(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNatKind(U22(z0))) → c61(ISNATKIND(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNatKind(U31(z0, z1))) → c61(ISNATKIND(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNatKind(U41(z0, z1, z2))) → c61(ISNATKIND(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNatKind(s(z0))) → c61(ISNATKIND(s(proper(z0))), PROPER(s(z0)))
PROPER(isNatKind(plus(z0, z1))) → c61(ISNATKIND(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNatKind(and(z0, z1))) → c61(ISNATKIND(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNatKind(isNatKind(z0))) → c61(ISNATKIND(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNatKind(tt)) → c61(ISNATKIND(ok(tt)))
PROPER(isNatKind(0)) → c61(ISNATKIND(ok(0)))
TOP(mark(U11(z0, z1, z2))) → c64(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(U12(z0, z1))) → c64(TOP(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
TOP(mark(isNat(z0))) → c64(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U13(z0))) → c64(TOP(U13(proper(z0))), PROPER(U13(z0)))
TOP(mark(U21(z0, z1))) → c64(TOP(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
TOP(mark(U22(z0))) → c64(TOP(U22(proper(z0))), PROPER(U22(z0)))
TOP(mark(U31(z0, z1))) → c64(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c64(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(s(z0))) → c64(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c64(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c64(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNatKind(z0))) → c64(TOP(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
TOP(mark(tt)) → c64(TOP(ok(tt)))
TOP(mark(0)) → c64(TOP(ok(0)))
TOP(ok(U11(tt, z0, z1))) → c65(TOP(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
TOP(ok(U12(tt, z0))) → c65(TOP(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
TOP(ok(U13(tt))) → c65(TOP(mark(tt)), ACTIVE(U13(tt)))
TOP(ok(U21(tt, z0))) → c65(TOP(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
TOP(ok(U22(tt))) → c65(TOP(mark(tt)), ACTIVE(U22(tt)))
TOP(ok(U31(tt, z0))) → c65(TOP(mark(z0)), ACTIVE(U31(tt, z0)))
TOP(ok(U41(tt, z0, z1))) → c65(TOP(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c65(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(0))) → c65(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(ok(isNat(plus(z0, z1)))) → c65(TOP(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c65(TOP(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
TOP(ok(isNatKind(0))) → c65(TOP(mark(tt)), ACTIVE(isNatKind(0)))
TOP(ok(isNatKind(plus(z0, z1)))) → c65(TOP(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
TOP(ok(isNatKind(s(z0)))) → c65(TOP(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
TOP(ok(plus(z0, 0))) → c65(TOP(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c65(TOP(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1, z2))) → c65(TOP(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
TOP(ok(U12(z0, z1))) → c65(TOP(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
TOP(ok(U13(z0))) → c65(TOP(U13(active(z0))), ACTIVE(U13(z0)))
TOP(ok(U21(z0, z1))) → c65(TOP(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
TOP(ok(U22(z0))) → c65(TOP(U22(active(z0))), ACTIVE(U22(z0)))
TOP(ok(U31(z0, z1))) → c65(TOP(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
TOP(ok(U41(z0, z1, z2))) → c65(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(s(z0))) → c65(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c65(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c65(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(and(z0, z1))) → c65(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND, PROPER, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55, c56, c56, c57, c57, c58, c58, c59, c59, c61, c61, c64, c64, c65
(119) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing nodes:
TOP(ok(isNatKind(0))) → c65(TOP(mark(tt)), ACTIVE(isNatKind(0)))
TOP(mark(tt)) → c64(TOP(ok(tt)))
TOP(ok(isNat(0))) → c65(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(ok(U13(tt))) → c65(TOP(mark(tt)), ACTIVE(U13(tt)))
TOP(ok(U22(tt))) → c65(TOP(mark(tt)), ACTIVE(U22(tt)))
TOP(mark(0)) → c64(TOP(ok(0)))
(120) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNatKind(U11(z0, z1, z2))) → c61(ISNATKIND(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNatKind(U12(z0, z1))) → c61(ISNATKIND(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNatKind(isNat(z0))) → c61(ISNATKIND(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNatKind(U13(z0))) → c61(ISNATKIND(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNatKind(U21(z0, z1))) → c61(ISNATKIND(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNatKind(U22(z0))) → c61(ISNATKIND(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNatKind(U31(z0, z1))) → c61(ISNATKIND(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNatKind(U41(z0, z1, z2))) → c61(ISNATKIND(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNatKind(s(z0))) → c61(ISNATKIND(s(proper(z0))), PROPER(s(z0)))
PROPER(isNatKind(plus(z0, z1))) → c61(ISNATKIND(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNatKind(and(z0, z1))) → c61(ISNATKIND(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNatKind(isNatKind(z0))) → c61(ISNATKIND(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNatKind(tt)) → c61(ISNATKIND(ok(tt)))
PROPER(isNatKind(0)) → c61(ISNATKIND(ok(0)))
TOP(mark(U11(z0, z1, z2))) → c64(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(U12(z0, z1))) → c64(TOP(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
TOP(mark(isNat(z0))) → c64(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U13(z0))) → c64(TOP(U13(proper(z0))), PROPER(U13(z0)))
TOP(mark(U21(z0, z1))) → c64(TOP(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
TOP(mark(U22(z0))) → c64(TOP(U22(proper(z0))), PROPER(U22(z0)))
TOP(mark(U31(z0, z1))) → c64(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c64(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(s(z0))) → c64(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c64(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c64(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNatKind(z0))) → c64(TOP(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
TOP(ok(U11(tt, z0, z1))) → c65(TOP(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
TOP(ok(U12(tt, z0))) → c65(TOP(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
TOP(ok(U21(tt, z0))) → c65(TOP(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
TOP(ok(U31(tt, z0))) → c65(TOP(mark(z0)), ACTIVE(U31(tt, z0)))
TOP(ok(U41(tt, z0, z1))) → c65(TOP(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c65(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(plus(z0, z1)))) → c65(TOP(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c65(TOP(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
TOP(ok(isNatKind(plus(z0, z1)))) → c65(TOP(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
TOP(ok(isNatKind(s(z0)))) → c65(TOP(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
TOP(ok(plus(z0, 0))) → c65(TOP(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c65(TOP(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1, z2))) → c65(TOP(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
TOP(ok(U12(z0, z1))) → c65(TOP(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
TOP(ok(U13(z0))) → c65(TOP(U13(active(z0))), ACTIVE(U13(z0)))
TOP(ok(U21(z0, z1))) → c65(TOP(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
TOP(ok(U22(z0))) → c65(TOP(U22(active(z0))), ACTIVE(U22(z0)))
TOP(ok(U31(z0, z1))) → c65(TOP(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
TOP(ok(U41(z0, z1, z2))) → c65(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(s(z0))) → c65(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c65(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c65(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(and(z0, z1))) → c65(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
S tuples:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNatKind(U11(z0, z1, z2))) → c61(ISNATKIND(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNatKind(U12(z0, z1))) → c61(ISNATKIND(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNatKind(isNat(z0))) → c61(ISNATKIND(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNatKind(U13(z0))) → c61(ISNATKIND(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNatKind(U21(z0, z1))) → c61(ISNATKIND(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNatKind(U22(z0))) → c61(ISNATKIND(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNatKind(U31(z0, z1))) → c61(ISNATKIND(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNatKind(U41(z0, z1, z2))) → c61(ISNATKIND(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNatKind(s(z0))) → c61(ISNATKIND(s(proper(z0))), PROPER(s(z0)))
PROPER(isNatKind(plus(z0, z1))) → c61(ISNATKIND(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNatKind(and(z0, z1))) → c61(ISNATKIND(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNatKind(isNatKind(z0))) → c61(ISNATKIND(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNatKind(tt)) → c61(ISNATKIND(ok(tt)))
PROPER(isNatKind(0)) → c61(ISNATKIND(ok(0)))
TOP(mark(U11(z0, z1, z2))) → c64(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(U12(z0, z1))) → c64(TOP(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
TOP(mark(isNat(z0))) → c64(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U13(z0))) → c64(TOP(U13(proper(z0))), PROPER(U13(z0)))
TOP(mark(U21(z0, z1))) → c64(TOP(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
TOP(mark(U22(z0))) → c64(TOP(U22(proper(z0))), PROPER(U22(z0)))
TOP(mark(U31(z0, z1))) → c64(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c64(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(s(z0))) → c64(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c64(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c64(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNatKind(z0))) → c64(TOP(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
TOP(ok(U11(tt, z0, z1))) → c65(TOP(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
TOP(ok(U12(tt, z0))) → c65(TOP(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
TOP(ok(U21(tt, z0))) → c65(TOP(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
TOP(ok(U31(tt, z0))) → c65(TOP(mark(z0)), ACTIVE(U31(tt, z0)))
TOP(ok(U41(tt, z0, z1))) → c65(TOP(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c65(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(plus(z0, z1)))) → c65(TOP(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c65(TOP(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
TOP(ok(isNatKind(plus(z0, z1)))) → c65(TOP(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
TOP(ok(isNatKind(s(z0)))) → c65(TOP(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
TOP(ok(plus(z0, 0))) → c65(TOP(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c65(TOP(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1, z2))) → c65(TOP(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
TOP(ok(U12(z0, z1))) → c65(TOP(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
TOP(ok(U13(z0))) → c65(TOP(U13(active(z0))), ACTIVE(U13(z0)))
TOP(ok(U21(z0, z1))) → c65(TOP(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
TOP(ok(U22(z0))) → c65(TOP(U22(active(z0))), ACTIVE(U22(z0)))
TOP(ok(U31(z0, z1))) → c65(TOP(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
TOP(ok(U41(z0, z1, z2))) → c65(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(s(z0))) → c65(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c65(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c65(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(and(z0, z1))) → c65(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
ACTIVE, U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND, PROPER, TOP
Compound Symbols:
c13, c24, c25, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63, c10, c2, c16, c16, c, c17, c17, c1, c18, c18, c3, c19, c19, c4, c20, c20, c5, c21, c21, c6, c22, c22, c7, c23, c23, c8, c26, c26, c9, c48, c48, c50, c50, c51, c51, c52, c52, c53, c53, c54, c54, c55, c55, c56, c56, c57, c57, c58, c58, c59, c59, c61, c61, c64, c65
(121) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(isNatKind(s(z0))) → c13(ISNATKIND(z0))
ACTIVE(plus(z0, z1)) → c24(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c25(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(isNat(s(z0))) → c10(ISNATKIND(z0))
ACTIVE(U11(tt, z0, z1)) → c2(U12'(isNat(z0), z1))
ACTIVE(U11(tt, z0, z1)) → c2(ISNAT(z0))
ACTIVE(U12(tt, z0)) → c2(U13'(isNat(z0)))
ACTIVE(U12(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U21(tt, z0)) → c2(U22'(isNat(z0)))
ACTIVE(U21(tt, z0)) → c2(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c2(S(plus(z1, z0)))
ACTIVE(U41(tt, z0, z1)) → c2(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c2(U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1))
ACTIVE(isNat(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNat(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(isNatKind(plus(z0, z1))) → c2(AND(isNatKind(z0), isNatKind(z1)))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z0))
ACTIVE(isNatKind(plus(z0, z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c2(U31'(and(isNat(z0), isNatKind(z0)), z0))
ACTIVE(plus(z0, 0)) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, 0)) → c2(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c2(ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z1))
ACTIVE(plus(z0, s(z1))) → c2(AND(isNat(z0), isNatKind(z0)))
ACTIVE(plus(z0, s(z1))) → c2(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c2(ISNATKIND(z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c16(U11'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c16(U11'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c16(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1), x1, x2)) → c16(U11'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U11(U13(z0), x1, x2)) → c16(U11'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U11(U21(z0, z1), x1, x2)) → c16(U11'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U11(U22(z0), x1, x2)) → c16(U11'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U11(U31(z0, z1), x1, x2)) → c16(U11'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1, x2)) → c16(U11'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c16(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c16(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1, x2)) → c16(U11'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U11(isNat(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(isNatKind(0), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(U11'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(U11'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U11(isNat(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(U11'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U11(isNat(s(z0)), x1, x2)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(U11'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U11(isNatKind(plus(z0, z1)), x1, x2)) → c(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(U11'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U11(isNatKind(s(z0)), x1, x2)) → c(ACTIVE(isNatKind(s(z0))))
ACTIVE(U12(plus(z0, 0), x1)) → c17(U12'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1)) → c17(U12'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1)) → c17(U12'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1), x1)) → c17(U12'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U12(U13(z0), x1)) → c17(U12'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U12(U21(z0, z1), x1)) → c17(U12'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U12(U22(z0), x1)) → c17(U12'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U12(U31(z0, z1), x1)) → c17(U12'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2), x1)) → c17(U12'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(s(z0), x1)) → c17(U12'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1)) → c17(U12'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U12(and(z0, z1), x1)) → c17(U12'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U12(isNat(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U12(isNatKind(0), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(U11'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U11(U12(tt, z0), x1, x2)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(U12'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U12(U11(tt, z0, z1), x1)) → c1(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0), x1)) → c1(U12'(mark(U13(isNat(z0))), x1))
ACTIVE(U12(U12(tt, z0), x1)) → c1(ACTIVE(U12(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(U12'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U12(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(U12'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U12(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(U12'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U12(isNatKind(plus(z0, z1)), x1)) → c1(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(U12'(mark(isNatKind(z0)), x1))
ACTIVE(U12(isNatKind(s(z0)), x1)) → c1(ACTIVE(isNatKind(s(z0))))
ACTIVE(U13(plus(z0, 0))) → c18(U13'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U13(plus(z0, s(z1)))) → c18(U13'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U13(U11(z0, z1, z2))) → c18(U13'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U13(U12(z0, z1))) → c18(U13'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U13(U13(z0))) → c18(U13'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U13(U21(z0, z1))) → c18(U13'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U13(U22(z0))) → c18(U13'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U13(U31(z0, z1))) → c18(U13'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U13(U41(z0, z1, z2))) → c18(U13'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U13(s(z0))) → c18(U13'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U13(plus(z0, z1))) → c18(U13'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U13(and(z0, z1))) → c18(U13'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U13(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U13(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U13(tt))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNat(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(isNatKind(0))) → c18(U13'(mark(tt)))
ACTIVE(U13(U11(tt, z0, z1))) → c3(U13'(mark(U12(isNat(z0), z1))))
ACTIVE(U13(U11(tt, z0, z1))) → c3(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U13(U12(tt, z0))) → c3(U13'(mark(U13(isNat(z0)))))
ACTIVE(U13(U12(tt, z0))) → c3(ACTIVE(U12(tt, z0)))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(U13'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U13(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U13(isNat(s(z0)))) → c3(U13'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U13(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(U13'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U13(isNatKind(plus(z0, z1)))) → c3(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(U13'(mark(isNatKind(z0))))
ACTIVE(U13(isNatKind(s(z0)))) → c3(ACTIVE(isNatKind(s(z0))))
ACTIVE(U21(plus(z0, 0), x1)) → c19(U21'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1)) → c19(U21'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1)) → c19(U21'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1), x1)) → c19(U21'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U21(U13(z0), x1)) → c19(U21'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U21(U21(z0, z1), x1)) → c19(U21'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U21(U22(z0), x1)) → c19(U21'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U21(U31(z0, z1), x1)) → c19(U21'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2), x1)) → c19(U21'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(s(z0), x1)) → c19(U21'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1)) → c19(U21'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1)) → c19(U21'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U21(U13(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNat(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U21(isNatKind(0), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(U11'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U11(U21(tt, z0), x1, x2)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U12(U21(tt, z0), x1)) → c4(U12'(mark(U22(isNat(z0))), x1))
ACTIVE(U12(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U13(U21(tt, z0))) → c4(U13'(mark(U22(isNat(z0)))))
ACTIVE(U13(U21(tt, z0))) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(U21'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U21(U11(tt, z0, z1), x1)) → c4(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0), x1)) → c4(U21'(mark(U13(isNat(z0))), x1))
ACTIVE(U21(U12(tt, z0), x1)) → c4(ACTIVE(U12(tt, z0)))
ACTIVE(U21(U21(tt, z0), x1)) → c4(U21'(mark(U22(isNat(z0))), x1))
ACTIVE(U21(U21(tt, z0), x1)) → c4(ACTIVE(U21(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(U21'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U21(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(U21'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U21(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(U21'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U21(isNatKind(plus(z0, z1)), x1)) → c4(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(U21'(mark(isNatKind(z0)), x1))
ACTIVE(U21(isNatKind(s(z0)), x1)) → c4(ACTIVE(isNatKind(s(z0))))
ACTIVE(U22(plus(z0, 0))) → c20(U22'(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)))) → c20(U22'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2))) → c20(U22'(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1))) → c20(U22'(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(U22(U13(z0))) → c20(U22'(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(U22(U21(z0, z1))) → c20(U22'(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(U22(U22(z0))) → c20(U22'(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(U22(U31(z0, z1))) → c20(U22'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U22(U41(z0, z1, z2))) → c20(U22'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U22(s(z0))) → c20(U22'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1))) → c20(U22'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U22(and(z0, z1))) → c20(U22'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U22(tt), x1, x2)) → c16(U11'(mark(tt), x1, x2))
ACTIVE(U12(U22(tt), x1)) → c17(U12'(mark(tt), x1))
ACTIVE(U13(U22(tt))) → c18(U13'(mark(tt)))
ACTIVE(U21(U22(tt), x1)) → c19(U21'(mark(tt), x1))
ACTIVE(U22(U13(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(U22(tt))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNat(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(isNatKind(0))) → c20(U22'(mark(tt)))
ACTIVE(U22(U11(tt, z0, z1))) → c5(U22'(mark(U12(isNat(z0), z1))))
ACTIVE(U22(U11(tt, z0, z1))) → c5(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0))) → c5(U22'(mark(U13(isNat(z0)))))
ACTIVE(U22(U12(tt, z0))) → c5(ACTIVE(U12(tt, z0)))
ACTIVE(U22(U21(tt, z0))) → c5(U22'(mark(U22(isNat(z0)))))
ACTIVE(U22(U21(tt, z0))) → c5(ACTIVE(U21(tt, z0)))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(U22'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(U22(isNat(plus(z0, z1)))) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U22(isNat(s(z0)))) → c5(U22'(mark(U21(isNatKind(z0), z0))))
ACTIVE(U22(isNat(s(z0)))) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(U22'(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(U22(isNatKind(plus(z0, z1)))) → c5(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(U22'(mark(isNatKind(z0))))
ACTIVE(U22(isNatKind(s(z0)))) → c5(ACTIVE(isNatKind(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c21(U31'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c21(U31'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1, z2), x1)) → c21(U31'(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U31(U12(z0, z1), x1)) → c21(U31'(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(U31(U13(z0), x1)) → c21(U31'(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(U31(U21(z0, z1), x1)) → c21(U31'(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(U31(U22(z0), x1)) → c21(U31'(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c21(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c21(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c21(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c21(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(and(z0, z1), x1)) → c21(U31'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(U31(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(U31(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(U31(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(U31(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(U13(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U22(tt), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(isNatKind(0), x1)) → c21(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(U31'(mark(U12(isNat(z0), z1)), x1))
ACTIVE(U31(U11(tt, z0, z1), x1)) → c6(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U31(U12(tt, z0), x1)) → c6(U31'(mark(U13(isNat(z0))), x1))
ACTIVE(U31(U12(tt, z0), x1)) → c6(ACTIVE(U12(tt, z0)))
ACTIVE(U31(U21(tt, z0), x1)) → c6(U31'(mark(U22(isNat(z0))), x1))
ACTIVE(U31(U21(tt, z0), x1)) → c6(ACTIVE(U21(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(U31'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(U31'(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(U31'(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(U31(isNatKind(plus(z0, z1)), x1)) → c6(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(U31'(mark(isNatKind(z0)), x1))
ACTIVE(U31(isNatKind(s(z0)), x1)) → c6(ACTIVE(isNatKind(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c22(U41'(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c22(U41'(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1, z2), x1, x2)) → c22(U41'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U41(U12(z0, z1), x1, x2)) → c22(U41'(U12(active(z0), z1), x1, x2), ACTIVE(U12(z0, z1)))
ACTIVE(U41(U13(z0), x1, x2)) → c22(U41'(U13(active(z0)), x1, x2), ACTIVE(U13(z0)))
ACTIVE(U41(U21(z0, z1), x1, x2)) → c22(U41'(U21(active(z0), z1), x1, x2), ACTIVE(U21(z0, z1)))
ACTIVE(U41(U22(z0), x1, x2)) → c22(U41'(U22(active(z0)), x1, x2), ACTIVE(U22(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c22(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c22(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c22(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c22(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c22(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U13(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U22(tt), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U41(isNatKind(0), x1, x2)) → c22(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(U12'(mark(s(plus(z1, z0))), x1))
ACTIVE(U12(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U13(U41(tt, z0, z1))) → c7(U13'(mark(s(plus(z1, z0)))))
ACTIVE(U13(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(U21'(mark(s(plus(z1, z0))), x1))
ACTIVE(U21(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U22(U41(tt, z0, z1))) → c7(U22'(mark(s(plus(z1, z0)))))
ACTIVE(U22(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(U41'(mark(U12(isNat(z0), z1)), x1, x2))
ACTIVE(U41(U11(tt, z0, z1), x1, x2)) → c7(ACTIVE(U11(tt, z0, z1)))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(U41'(mark(U13(isNat(z0))), x1, x2))
ACTIVE(U41(U12(tt, z0), x1, x2)) → c7(ACTIVE(U12(tt, z0)))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(U41'(mark(U22(isNat(z0))), x1, x2))
ACTIVE(U41(U21(tt, z0), x1, x2)) → c7(ACTIVE(U21(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(U41'(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(U41'(mark(U21(isNatKind(z0), z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c7(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(U41'(mark(and(isNatKind(z0), isNatKind(z1))), x1, x2))
ACTIVE(U41(isNatKind(plus(z0, z1)), x1, x2)) → c7(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(U41'(mark(isNatKind(z0)), x1, x2))
ACTIVE(U41(isNatKind(s(z0)), x1, x2)) → c7(ACTIVE(isNatKind(s(z0))))
ACTIVE(s(plus(z0, 0))) → c23(S(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c23(S(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c23(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1))) → c23(S(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
ACTIVE(s(U13(z0))) → c23(S(U13(active(z0))), ACTIVE(U13(z0)))
ACTIVE(s(U21(z0, z1))) → c23(S(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
ACTIVE(s(U22(z0))) → c23(S(U22(active(z0))), ACTIVE(U22(z0)))
ACTIVE(s(U31(z0, z1))) → c23(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c23(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(s(z0))) → c23(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c23(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c23(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U13(tt))) → c23(S(mark(tt)))
ACTIVE(s(U22(tt))) → c23(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c23(S(mark(z0)))
ACTIVE(s(isNat(0))) → c23(S(mark(tt)))
ACTIVE(s(isNatKind(0))) → c23(S(mark(tt)))
ACTIVE(s(U11(tt, z0, z1))) → c8(S(mark(U12(isNat(z0), z1))))
ACTIVE(s(U11(tt, z0, z1))) → c8(ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0))) → c8(S(mark(U13(isNat(z0)))))
ACTIVE(s(U12(tt, z0))) → c8(ACTIVE(U12(tt, z0)))
ACTIVE(s(U21(tt, z0))) → c8(S(mark(U22(isNat(z0)))))
ACTIVE(s(U21(tt, z0))) → c8(ACTIVE(U21(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c8(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U41(tt, z0, z1))) → c8(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(S(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c8(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c8(S(mark(U21(isNatKind(z0), z0))))
ACTIVE(s(isNat(s(z0)))) → c8(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(S(mark(and(isNatKind(z0), isNatKind(z1)))))
ACTIVE(s(isNatKind(plus(z0, z1)))) → c8(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(s(isNatKind(s(z0)))) → c8(S(mark(isNatKind(z0))))
ACTIVE(s(isNatKind(s(z0)))) → c8(ACTIVE(isNatKind(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c26(AND(mark(U31(and(isNat(z0), isNatKind(z0)), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c26(AND(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1, z2), x1)) → c26(AND(U11(active(z0), z1, z2), x1), ACTIVE(U11(z0, z1, z2)))
ACTIVE(and(U12(z0, z1), x1)) → c26(AND(U12(active(z0), z1), x1), ACTIVE(U12(z0, z1)))
ACTIVE(and(U13(z0), x1)) → c26(AND(U13(active(z0)), x1), ACTIVE(U13(z0)))
ACTIVE(and(U21(z0, z1), x1)) → c26(AND(U21(active(z0), z1), x1), ACTIVE(U21(z0, z1)))
ACTIVE(and(U22(z0), x1)) → c26(AND(U22(active(z0)), x1), ACTIVE(U22(z0)))
ACTIVE(and(U31(z0, z1), x1)) → c26(AND(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c26(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c26(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c26(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c26(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1, x2)) → c16(U11'(mark(z0), x1, x2))
ACTIVE(U12(and(tt, z0), x1)) → c17(U12'(mark(z0), x1))
ACTIVE(U13(and(tt, z0))) → c18(U13'(mark(z0)))
ACTIVE(U21(and(tt, z0), x1)) → c19(U21'(mark(z0), x1))
ACTIVE(U22(and(tt, z0))) → c20(U22'(mark(z0)))
ACTIVE(U31(and(tt, z0), x1)) → c21(U31'(mark(z0), x1))
ACTIVE(U41(and(tt, z0), x1, x2)) → c22(U41'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c23(S(mark(z0)))
ACTIVE(and(U13(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U22(tt), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U31(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c26(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(isNatKind(0), x1)) → c26(AND(mark(tt), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(AND(mark(U12(isNat(z0), z1)), x1))
ACTIVE(and(U11(tt, z0, z1), x1)) → c9(ACTIVE(U11(tt, z0, z1)))
ACTIVE(and(U12(tt, z0), x1)) → c9(AND(mark(U13(isNat(z0))), x1))
ACTIVE(and(U12(tt, z0), x1)) → c9(ACTIVE(U12(tt, z0)))
ACTIVE(and(U21(tt, z0), x1)) → c9(AND(mark(U22(isNat(z0))), x1))
ACTIVE(and(U21(tt, z0), x1)) → c9(ACTIVE(U21(tt, z0)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c9(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(AND(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c9(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c9(AND(mark(U21(isNatKind(z0), z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c9(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(AND(mark(and(isNatKind(z0), isNatKind(z1))), x1))
ACTIVE(and(isNatKind(plus(z0, z1)), x1)) → c9(ACTIVE(isNatKind(plus(z0, z1))))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(AND(mark(isNatKind(z0)), x1))
ACTIVE(and(isNatKind(s(z0)), x1)) → c9(ACTIVE(isNatKind(s(z0))))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1))) → c48(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U11(x0, x1, isNat(z0))) → c48(U11'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U11(x0, x1, U13(z0))) → c48(U11'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U11(x0, x1, U21(z0, z1))) → c48(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U11(x0, x1, U22(z0))) → c48(U11'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U11(x0, x1, U31(z0, z1))) → c48(U11'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U11(x0, x1, U41(z0, z1, z2))) → c48(U11'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c48(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c48(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, and(z0, z1))) → c48(U11'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U11(x0, x1, isNatKind(z0))) → c48(U11'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c48(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1), x2)) → c48(U11'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNat(z0), x2)) → c48(U11'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U11(x0, U13(z0), x2)) → c48(U11'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1), x2)) → c48(U11'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U11(x0, U22(z0), x2)) → c48(U11'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U11(x0, U31(z0, z1), x2)) → c48(U11'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U11(x0, U41(z0, z1, z2), x2)) → c48(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c48(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c48(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, and(z0, z1), x2)) → c48(U11'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U11(x0, isNatKind(z0), x2)) → c48(U11'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c48(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1), x1, x2)) → c48(U11'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNat(z0), x1, x2)) → c48(U11'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U13(z0), x1, x2)) → c48(U11'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1), x1, x2)) → c48(U11'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0), x1, x2)) → c48(U11'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(U31(z0, z1), x1, x2)) → c48(U11'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U41(z0, z1, z2), x1, x2)) → c48(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c48(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c48(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(and(z0, z1), x1, x2)) → c48(U11'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(isNatKind(z0), x1, x2)) → c48(U11'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c48(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c48(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c48(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c48(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c48(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c48(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, U11(z0, z1, z2))) → c50(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, U12(z0, z1))) → c50(U12'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U12(x0, isNat(z0))) → c50(U12'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U12(x0, U13(z0))) → c50(U12'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U12(x0, U21(z0, z1))) → c50(U12'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U12(x0, U22(z0))) → c50(U12'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U12(x0, U31(z0, z1))) → c50(U12'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U12(x0, U41(z0, z1, z2))) → c50(U12'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U12(x0, s(z0))) → c50(U12'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U12(x0, plus(z0, z1))) → c50(U12'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U12(x0, and(z0, z1))) → c50(U12'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U12(x0, isNatKind(z0))) → c50(U12'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U12(U11(z0, z1, z2), x1)) → c50(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U12(U12(z0, z1), x1)) → c50(U12'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U12(isNat(z0), x1)) → c50(U12'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U12(U13(z0), x1)) → c50(U12'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U12(U21(z0, z1), x1)) → c50(U12'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U12(U22(z0), x1)) → c50(U12'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U12(U31(z0, z1), x1)) → c50(U12'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U12(U41(z0, z1, z2), x1)) → c50(U12'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U12(s(z0), x1)) → c50(U12'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U12(plus(z0, z1), x1)) → c50(U12'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U12(and(z0, z1), x1)) → c50(U12'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U12(isNatKind(z0), x1)) → c50(U12'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U12(x0, tt)) → c50(U12'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U12(x0, 0)) → c50(U12'(proper(x0), ok(0)), PROPER(x0))
PROPER(U12(tt, x1)) → c50(U12'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U12(0, x1)) → c50(U12'(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1, z2))) → c51(ISNAT(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNat(U12(z0, z1))) → c51(ISNAT(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U13(z0))) → c51(ISNAT(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNat(U21(z0, z1))) → c51(ISNAT(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNat(U22(z0))) → c51(ISNAT(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNat(U31(z0, z1))) → c51(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNatKind(z0))) → c51(ISNAT(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
PROPER(U13(U11(z0, z1, z2))) → c52(U13'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U13(U12(z0, z1))) → c52(U13'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U13(isNat(z0))) → c52(U13'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U13(U13(z0))) → c52(U13'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U13(U21(z0, z1))) → c52(U13'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U13(U22(z0))) → c52(U13'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U13(U31(z0, z1))) → c52(U13'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U13(U41(z0, z1, z2))) → c52(U13'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U13(s(z0))) → c52(U13'(s(proper(z0))), PROPER(s(z0)))
PROPER(U13(plus(z0, z1))) → c52(U13'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U13(and(z0, z1))) → c52(U13'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U13(isNatKind(z0))) → c52(U13'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U13(tt)) → c52(U13'(ok(tt)))
PROPER(U13(0)) → c52(U13'(ok(0)))
PROPER(U21(x0, U11(z0, z1, z2))) → c53(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, U12(z0, z1))) → c53(U21'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U21(x0, isNat(z0))) → c53(U21'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U21(x0, U13(z0))) → c53(U21'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U21(x0, U21(z0, z1))) → c53(U21'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U21(x0, U22(z0))) → c53(U21'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U21(x0, U31(z0, z1))) → c53(U21'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U21(x0, U41(z0, z1, z2))) → c53(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, s(z0))) → c53(U21'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U21(x0, plus(z0, z1))) → c53(U21'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U21(x0, and(z0, z1))) → c53(U21'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U21(x0, isNatKind(z0))) → c53(U21'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U21(U11(z0, z1, z2), x1)) → c53(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U21(U12(z0, z1), x1)) → c53(U21'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U21(isNat(z0), x1)) → c53(U21'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U21(U13(z0), x1)) → c53(U21'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U21(U21(z0, z1), x1)) → c53(U21'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U21(U22(z0), x1)) → c53(U21'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U21(U31(z0, z1), x1)) → c53(U21'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U21(U41(z0, z1, z2), x1)) → c53(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U21(s(z0), x1)) → c53(U21'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U21(plus(z0, z1), x1)) → c53(U21'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U21(and(z0, z1), x1)) → c53(U21'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U21(isNatKind(z0), x1)) → c53(U21'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U21(x0, tt)) → c53(U21'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U21(x0, 0)) → c53(U21'(proper(x0), ok(0)), PROPER(x0))
PROPER(U21(tt, x1)) → c53(U21'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U21(0, x1)) → c53(U21'(ok(0), proper(x1)), PROPER(x1))
PROPER(U22(U11(z0, z1, z2))) → c54(U22'(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(U22(U12(z0, z1))) → c54(U22'(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(U22(isNat(z0))) → c54(U22'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U22(U13(z0))) → c54(U22'(U13(proper(z0))), PROPER(U13(z0)))
PROPER(U22(U21(z0, z1))) → c54(U22'(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(U22(U22(z0))) → c54(U22'(U22(proper(z0))), PROPER(U22(z0)))
PROPER(U22(U31(z0, z1))) → c54(U22'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U22(U41(z0, z1, z2))) → c54(U22'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U22(s(z0))) → c54(U22'(s(proper(z0))), PROPER(s(z0)))
PROPER(U22(plus(z0, z1))) → c54(U22'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U22(and(z0, z1))) → c54(U22'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U22(isNatKind(z0))) → c54(U22'(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(U22(tt)) → c54(U22'(ok(tt)))
PROPER(U22(0)) → c54(U22'(ok(0)))
PROPER(U31(x0, U11(z0, z1, z2))) → c55(U31'(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(U31(x0, U12(z0, z1))) → c55(U31'(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(U31(x0, isNat(z0))) → c55(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U13(z0))) → c55(U31'(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(U31(x0, U21(z0, z1))) → c55(U31'(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(U31(x0, U22(z0))) → c55(U31'(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(U31(x0, U31(z0, z1))) → c55(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c55(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c55(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c55(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, and(z0, z1))) → c55(U31'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U31(x0, isNatKind(z0))) → c55(U31'(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(U31(U11(z0, z1, z2), x1)) → c55(U31'(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(U31(U12(z0, z1), x1)) → c55(U31'(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c55(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U13(z0), x1)) → c55(U31'(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(U31(U21(z0, z1), x1)) → c55(U31'(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(U31(U22(z0), x1)) → c55(U31'(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c55(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c55(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c55(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c55(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(and(z0, z1), x1)) → c55(U31'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U31(isNatKind(z0), x1)) → c55(U31'(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(U31(x0, tt)) → c55(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c55(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c55(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c55(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U41(x0, x1, U12(z0, z1))) → c56(U41'(proper(x0), proper(x1), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c56(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U13(z0))) → c56(U41'(proper(x0), proper(x1), U13(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U13(z0)))
PROPER(U41(x0, x1, U21(z0, z1))) → c56(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1)))
PROPER(U41(x0, x1, U22(z0))) → c56(U41'(proper(x0), proper(x1), U22(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U22(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c56(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c56(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c56(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c56(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c56(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNatKind(z0))) → c56(U41'(proper(x0), proper(x1), isNatKind(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNatKind(z0)))
PROPER(U41(x0, U11(z0, z1, z2), x2)) → c56(U41'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U12(z0, z1), x2)) → c56(U41'(proper(x0), U12(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c56(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U13(z0), x2)) → c56(U41'(proper(x0), U13(proper(z0)), proper(x2)), PROPER(x0), PROPER(U13(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1), x2)) → c56(U41'(proper(x0), U21(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1)), PROPER(x2))
PROPER(U41(x0, U22(z0), x2)) → c56(U41'(proper(x0), U22(proper(z0)), proper(x2)), PROPER(x0), PROPER(U22(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c56(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c56(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c56(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c56(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c56(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNatKind(z0), x2)) → c56(U41'(proper(x0), isNatKind(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNatKind(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1, z2), x1, x2)) → c56(U41'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0, z1), x1, x2)) → c56(U41'(U12(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U12(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c56(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U13(z0), x1, x2)) → c56(U41'(U13(proper(z0)), proper(x1), proper(x2)), PROPER(U13(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1), x1, x2)) → c56(U41'(U21(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U21(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U22(z0), x1, x2)) → c56(U41'(U22(proper(z0)), proper(x1), proper(x2)), PROPER(U22(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c56(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c56(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c56(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c56(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c56(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNatKind(z0), x1, x2)) → c56(U41'(isNatKind(proper(z0)), proper(x1), proper(x2)), PROPER(isNatKind(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c56(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c56(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c56(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c56(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c56(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c56(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c57(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1))) → c57(S(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(s(isNat(z0))) → c57(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U13(z0))) → c57(S(U13(proper(z0))), PROPER(U13(z0)))
PROPER(s(U21(z0, z1))) → c57(S(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(s(U22(z0))) → c57(S(U22(proper(z0))), PROPER(U22(z0)))
PROPER(s(U31(z0, z1))) → c57(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c57(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(s(z0))) → c57(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c57(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c57(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNatKind(z0))) → c57(S(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(s(tt)) → c57(S(ok(tt)))
PROPER(s(0)) → c57(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c58(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1))) → c58(PLUS(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c58(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U13(z0))) → c58(PLUS(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(plus(x0, U21(z0, z1))) → c58(PLUS(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(plus(x0, U22(z0))) → c58(PLUS(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(plus(x0, U31(z0, z1))) → c58(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c58(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c58(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c58(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c58(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNatKind(z0))) → c58(PLUS(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(plus(U11(z0, z1, z2), x1)) → c58(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1), x1)) → c58(PLUS(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c58(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U13(z0), x1)) → c58(PLUS(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(plus(U21(z0, z1), x1)) → c58(PLUS(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(plus(U22(z0), x1)) → c58(PLUS(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c58(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c58(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c58(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c58(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c58(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNatKind(z0), x1)) → c58(PLUS(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c58(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c58(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c58(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c58(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1, z2))) → c59(AND(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(and(x0, U12(z0, z1))) → c59(AND(proper(x0), U12(proper(z0), proper(z1))), PROPER(x0), PROPER(U12(z0, z1)))
PROPER(and(x0, isNat(z0))) → c59(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, U13(z0))) → c59(AND(proper(x0), U13(proper(z0))), PROPER(x0), PROPER(U13(z0)))
PROPER(and(x0, U21(z0, z1))) → c59(AND(proper(x0), U21(proper(z0), proper(z1))), PROPER(x0), PROPER(U21(z0, z1)))
PROPER(and(x0, U22(z0))) → c59(AND(proper(x0), U22(proper(z0))), PROPER(x0), PROPER(U22(z0)))
PROPER(and(x0, U31(z0, z1))) → c59(AND(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(and(x0, U41(z0, z1, z2))) → c59(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c59(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c59(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c59(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNatKind(z0))) → c59(AND(proper(x0), isNatKind(proper(z0))), PROPER(x0), PROPER(isNatKind(z0)))
PROPER(and(U11(z0, z1, z2), x1)) → c59(AND(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(and(U12(z0, z1), x1)) → c59(AND(U12(proper(z0), proper(z1)), proper(x1)), PROPER(U12(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c59(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(U13(z0), x1)) → c59(AND(U13(proper(z0)), proper(x1)), PROPER(U13(z0)), PROPER(x1))
PROPER(and(U21(z0, z1), x1)) → c59(AND(U21(proper(z0), proper(z1)), proper(x1)), PROPER(U21(z0, z1)), PROPER(x1))
PROPER(and(U22(z0), x1)) → c59(AND(U22(proper(z0)), proper(x1)), PROPER(U22(z0)), PROPER(x1))
PROPER(and(U31(z0, z1), x1)) → c59(AND(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c59(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c59(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c59(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c59(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNatKind(z0), x1)) → c59(AND(isNatKind(proper(z0)), proper(x1)), PROPER(isNatKind(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c59(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c59(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c59(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c59(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNatKind(U11(z0, z1, z2))) → c61(ISNATKIND(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(isNatKind(U12(z0, z1))) → c61(ISNATKIND(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
PROPER(isNatKind(isNat(z0))) → c61(ISNATKIND(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNatKind(U13(z0))) → c61(ISNATKIND(U13(proper(z0))), PROPER(U13(z0)))
PROPER(isNatKind(U21(z0, z1))) → c61(ISNATKIND(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
PROPER(isNatKind(U22(z0))) → c61(ISNATKIND(U22(proper(z0))), PROPER(U22(z0)))
PROPER(isNatKind(U31(z0, z1))) → c61(ISNATKIND(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNatKind(U41(z0, z1, z2))) → c61(ISNATKIND(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNatKind(s(z0))) → c61(ISNATKIND(s(proper(z0))), PROPER(s(z0)))
PROPER(isNatKind(plus(z0, z1))) → c61(ISNATKIND(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNatKind(and(z0, z1))) → c61(ISNATKIND(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNatKind(isNatKind(z0))) → c61(ISNATKIND(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
PROPER(isNatKind(tt)) → c61(ISNATKIND(ok(tt)))
PROPER(isNatKind(0)) → c61(ISNATKIND(ok(0)))
TOP(mark(U11(z0, z1, z2))) → c64(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(U12(z0, z1))) → c64(TOP(U12(proper(z0), proper(z1))), PROPER(U12(z0, z1)))
TOP(mark(isNat(z0))) → c64(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U13(z0))) → c64(TOP(U13(proper(z0))), PROPER(U13(z0)))
TOP(mark(U21(z0, z1))) → c64(TOP(U21(proper(z0), proper(z1))), PROPER(U21(z0, z1)))
TOP(mark(U22(z0))) → c64(TOP(U22(proper(z0))), PROPER(U22(z0)))
TOP(mark(U31(z0, z1))) → c64(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c64(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(s(z0))) → c64(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c64(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c64(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNatKind(z0))) → c64(TOP(isNatKind(proper(z0))), PROPER(isNatKind(z0)))
TOP(ok(U11(tt, z0, z1))) → c65(TOP(mark(U12(isNat(z0), z1))), ACTIVE(U11(tt, z0, z1)))
TOP(ok(U12(tt, z0))) → c65(TOP(mark(U13(isNat(z0)))), ACTIVE(U12(tt, z0)))
TOP(ok(U21(tt, z0))) → c65(TOP(mark(U22(isNat(z0)))), ACTIVE(U21(tt, z0)))
TOP(ok(U31(tt, z0))) → c65(TOP(mark(z0)), ACTIVE(U31(tt, z0)))
TOP(ok(U41(tt, z0, z1))) → c65(TOP(mark(s(plus(z1, z0)))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c65(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(plus(z0, z1)))) → c65(TOP(mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c65(TOP(mark(U21(isNatKind(z0), z0))), ACTIVE(isNat(s(z0))))
TOP(ok(isNatKind(plus(z0, z1)))) → c65(TOP(mark(and(isNatKind(z0), isNatKind(z1)))), ACTIVE(isNatKind(plus(z0, z1))))
TOP(ok(isNatKind(s(z0)))) → c65(TOP(mark(isNatKind(z0))), ACTIVE(isNatKind(s(z0))))
TOP(ok(plus(z0, 0))) → c65(TOP(mark(U31(and(isNat(z0), isNatKind(z0)), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c65(TOP(mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1, z2))) → c65(TOP(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
TOP(ok(U12(z0, z1))) → c65(TOP(U12(active(z0), z1)), ACTIVE(U12(z0, z1)))
TOP(ok(U13(z0))) → c65(TOP(U13(active(z0))), ACTIVE(U13(z0)))
TOP(ok(U21(z0, z1))) → c65(TOP(U21(active(z0), z1)), ACTIVE(U21(z0, z1)))
TOP(ok(U22(z0))) → c65(TOP(U22(active(z0))), ACTIVE(U22(z0)))
TOP(ok(U31(z0, z1))) → c65(TOP(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
TOP(ok(U41(z0, z1, z2))) → c65(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(s(z0))) → c65(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c65(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c65(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(and(z0, z1))) → c65(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
(122) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
S tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
K tuples:none
Defined Rule Symbols:
active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U31, U21, U41, proper
Defined Pair Symbols:
U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND
Compound Symbols:
c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63
(123) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1)) → U12(active(z0), z1)
active(U13(z0)) → U13(active(z0))
active(U21(z0, z1)) → U21(active(z0), z1)
active(U22(z0)) → U22(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U12(mark(z0), z1) → mark(U12(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U13(mark(z0)) → mark(U13(z0))
U13(ok(z0)) → ok(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1)) → U12(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(U13(z0)) → U13(proper(z0))
proper(U21(z0, z1)) → U21(proper(z0), proper(z1))
proper(U22(z0)) → U22(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(0) → ok(0)
proper(isNatKind(z0)) → isNatKind(proper(z0))
(124) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
S tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND
Compound Symbols:
c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63
(125) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AND(x1, x2)) = 0
POL(ISNAT(x1)) = 0
POL(ISNATKIND(x1)) = 0
POL(PLUS(x1, x2)) = 0
POL(S(x1)) = 0
POL(U11'(x1, x2, x3)) = 0
POL(U12'(x1, x2)) = [2]x1
POL(U13'(x1)) = 0
POL(U21'(x1, x2)) = 0
POL(U22'(x1)) = 0
POL(U31'(x1, x2)) = 0
POL(U41'(x1, x2, x3)) = 0
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(c44(x1)) = x1
POL(c45(x1)) = x1
POL(c46(x1)) = x1
POL(c47(x1)) = x1
POL(c62(x1)) = x1
POL(c63(x1)) = x1
POL(mark(x1)) = x1
POL(ok(x1)) = [1] + x1
(126) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
S tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
K tuples:
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND
Compound Symbols:
c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63
(127) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AND(x1, x2)) = 0
POL(ISNAT(x1)) = 0
POL(ISNATKIND(x1)) = 0
POL(PLUS(x1, x2)) = [2]x2
POL(S(x1)) = [4]x1
POL(U11'(x1, x2, x3)) = 0
POL(U12'(x1, x2)) = [4]x2
POL(U13'(x1)) = 0
POL(U21'(x1, x2)) = x1
POL(U22'(x1)) = [4]x1
POL(U31'(x1, x2)) = 0
POL(U41'(x1, x2, x3)) = [3]x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(c44(x1)) = x1
POL(c45(x1)) = x1
POL(c46(x1)) = x1
POL(c47(x1)) = x1
POL(c62(x1)) = x1
POL(c63(x1)) = x1
POL(mark(x1)) = [4] + x1
POL(ok(x1)) = [4] + x1
(128) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
S tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
K tuples:
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND
Compound Symbols:
c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63
(129) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
U31'(mark(z0), z1) → c37(U31'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AND(x1, x2)) = 0
POL(ISNAT(x1)) = 0
POL(ISNATKIND(x1)) = 0
POL(PLUS(x1, x2)) = [4]x2
POL(S(x1)) = [2]x1
POL(U11'(x1, x2, x3)) = 0
POL(U12'(x1, x2)) = 0
POL(U13'(x1)) = 0
POL(U21'(x1, x2)) = [4]x1
POL(U22'(x1)) = [4]x1
POL(U31'(x1, x2)) = [2]x1
POL(U41'(x1, x2, x3)) = x2 + [5]x3
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(c44(x1)) = x1
POL(c45(x1)) = x1
POL(c46(x1)) = x1
POL(c47(x1)) = x1
POL(c62(x1)) = x1
POL(c63(x1)) = x1
POL(mark(x1)) = [1] + x1
POL(ok(x1)) = x1
(130) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
S tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
K tuples:
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND
Compound Symbols:
c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63
(131) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AND(x1, x2)) = 0
POL(ISNAT(x1)) = [4]x1
POL(ISNATKIND(x1)) = 0
POL(PLUS(x1, x2)) = 0
POL(S(x1)) = 0
POL(U11'(x1, x2, x3)) = [4]x2 + [4]x3
POL(U12'(x1, x2)) = [2]x2
POL(U13'(x1)) = 0
POL(U21'(x1, x2)) = 0
POL(U22'(x1)) = 0
POL(U31'(x1, x2)) = [4]x2
POL(U41'(x1, x2, x3)) = 0
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(c44(x1)) = x1
POL(c45(x1)) = x1
POL(c46(x1)) = x1
POL(c47(x1)) = x1
POL(c62(x1)) = x1
POL(c63(x1)) = x1
POL(mark(x1)) = 0
POL(ok(x1)) = [4] + x1
(132) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
S tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
K tuples:
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND
Compound Symbols:
c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63
(133) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AND(x1, x2)) = 0
POL(ISNAT(x1)) = [3]x1
POL(ISNATKIND(x1)) = [4]x1
POL(PLUS(x1, x2)) = 0
POL(S(x1)) = 0
POL(U11'(x1, x2, x3)) = [3]x2 + [4]x3
POL(U12'(x1, x2)) = [2]x2
POL(U13'(x1)) = 0
POL(U21'(x1, x2)) = 0
POL(U22'(x1)) = 0
POL(U31'(x1, x2)) = 0
POL(U41'(x1, x2, x3)) = [2]x2 + [4]x3
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(c44(x1)) = x1
POL(c45(x1)) = x1
POL(c46(x1)) = x1
POL(c47(x1)) = x1
POL(c62(x1)) = x1
POL(c63(x1)) = x1
POL(mark(x1)) = [5]
POL(ok(x1)) = [4] + x1
(134) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
S tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
K tuples:
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND
Compound Symbols:
c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63
(135) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AND(x1, x2)) = x2
POL(ISNAT(x1)) = [4]x1
POL(ISNATKIND(x1)) = [3]x1
POL(PLUS(x1, x2)) = 0
POL(S(x1)) = 0
POL(U11'(x1, x2, x3)) = [2]x2 + [2]x3
POL(U12'(x1, x2)) = 0
POL(U13'(x1)) = 0
POL(U21'(x1, x2)) = 0
POL(U22'(x1)) = 0
POL(U31'(x1, x2)) = 0
POL(U41'(x1, x2, x3)) = [4]x2
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(c44(x1)) = x1
POL(c45(x1)) = x1
POL(c46(x1)) = x1
POL(c47(x1)) = x1
POL(c62(x1)) = x1
POL(c63(x1)) = x1
POL(mark(x1)) = [4]
POL(ok(x1)) = [1] + x1
(136) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
S tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
K tuples:
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND
Compound Symbols:
c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63
(137) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AND(x1, x2)) = [2]x2
POL(ISNAT(x1)) = [4]x1
POL(ISNATKIND(x1)) = 0
POL(PLUS(x1, x2)) = x1 + [4]x2
POL(S(x1)) = 0
POL(U11'(x1, x2, x3)) = x1 + [3]x2 + [4]x3
POL(U12'(x1, x2)) = [2]x1
POL(U13'(x1)) = [5]x1
POL(U21'(x1, x2)) = 0
POL(U22'(x1)) = [4]x1
POL(U31'(x1, x2)) = [2]x1
POL(U41'(x1, x2, x3)) = [5]x1 + [2]x2 + [5]x3
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(c44(x1)) = x1
POL(c45(x1)) = x1
POL(c46(x1)) = x1
POL(c47(x1)) = x1
POL(c62(x1)) = x1
POL(c63(x1)) = x1
POL(mark(x1)) = [5] + x1
POL(ok(x1)) = x1
(138) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
S tuples:
U13'(ok(z0)) → c32(U13'(z0))
AND(mark(z0), z1) → c46(AND(z0, z1))
K tuples:
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND
Compound Symbols:
c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63
(139) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
AND(mark(z0), z1) → c46(AND(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AND(x1, x2)) = [4]x1 + [2]x2
POL(ISNAT(x1)) = [4]x1
POL(ISNATKIND(x1)) = [2]x1
POL(PLUS(x1, x2)) = [2]x1 + [4]x2
POL(S(x1)) = [2]x1
POL(U11'(x1, x2, x3)) = [3]x2 + [3]x3
POL(U12'(x1, x2)) = [3]x2
POL(U13'(x1)) = 0
POL(U21'(x1, x2)) = [4]x2
POL(U22'(x1)) = [2]x1
POL(U31'(x1, x2)) = [5]x2
POL(U41'(x1, x2, x3)) = [4]x1 + [2]x3
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(c44(x1)) = x1
POL(c45(x1)) = x1
POL(c46(x1)) = x1
POL(c47(x1)) = x1
POL(c62(x1)) = x1
POL(c63(x1)) = x1
POL(mark(x1)) = [2] + x1
POL(ok(x1)) = [4] + x1
(140) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
S tuples:
U13'(ok(z0)) → c32(U13'(z0))
K tuples:
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND
Compound Symbols:
c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63
(141) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
U13'(ok(z0)) → c32(U13'(z0))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AND(x1, x2)) = [2]x1
POL(ISNAT(x1)) = 0
POL(ISNATKIND(x1)) = 0
POL(PLUS(x1, x2)) = [3]x1 + [2]x2
POL(S(x1)) = [4]x1
POL(U11'(x1, x2, x3)) = [2]x2 + x3
POL(U12'(x1, x2)) = [5]x2
POL(U13'(x1)) = x1
POL(U21'(x1, x2)) = [4]x1
POL(U22'(x1)) = 0
POL(U31'(x1, x2)) = [2]x1 + [2]x2
POL(U41'(x1, x2, x3)) = [2]x2 + [2]x3
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(c44(x1)) = x1
POL(c45(x1)) = x1
POL(c46(x1)) = x1
POL(c47(x1)) = x1
POL(c62(x1)) = x1
POL(c63(x1)) = x1
POL(mark(x1)) = x1
POL(ok(x1)) = [1] + x1
(142) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
U13'(ok(z0)) → c32(U13'(z0))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
S tuples:none
K tuples:
U12'(ok(z0), ok(z1)) → c30(U12'(z0, z1))
U21'(mark(z0), z1) → c33(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c34(U21'(z0, z1))
U22'(mark(z0)) → c35(U22'(z0))
U22'(ok(z0)) → c36(U22'(z0))
U41'(mark(z0), z1, z2) → c39(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c40(U41'(z0, z1, z2))
S(mark(z0)) → c41(S(z0))
S(ok(z0)) → c42(S(z0))
PLUS(z0, mark(z1)) → c44(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c45(PLUS(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
ISNAT(ok(z0)) → c62(ISNAT(z0))
ISNATKIND(ok(z0)) → c63(ISNATKIND(z0))
AND(ok(z0), ok(z1)) → c47(AND(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c29(U12'(z0, z1))
U13'(mark(z0)) → c31(U13'(z0))
PLUS(mark(z0), z1) → c43(PLUS(z0, z1))
AND(mark(z0), z1) → c46(AND(z0, z1))
U13'(ok(z0)) → c32(U13'(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', U13', U21', U22', U31', U41', S, PLUS, AND, ISNAT, ISNATKIND
Compound Symbols:
c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c62, c63
(143) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(144) BOUNDS(O(1), O(1))