(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,