(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2, X3)) → U22(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2, X3) → mark(U22(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3)) → U22(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(0) → ok(0)
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3)) → ok(U22(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(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))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(U11(tt, z0, z1)) → c(U12'(tt, z0, z1))
ACTIVE(U12(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U21(tt, z0, z1)) → c2(U22'(tt, z0, z1))
ACTIVE(U22(tt, z0, z1)) → c3(PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(plus(z0, 0)) → c4
ACTIVE(plus(z0, s(z1))) → c5(U11'(tt, z1, z0))
ACTIVE(x(z0, 0)) → c6
ACTIVE(x(z0, s(z1))) → c7(U21'(tt, z1, z0))
ACTIVE(U11(z0, z1, z2)) → c8(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c9(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(tt) → c34
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c41
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
S tuples:
ACTIVE(U11(tt, z0, z1)) → c(U12'(tt, z0, z1))
ACTIVE(U12(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U21(tt, z0, z1)) → c2(U22'(tt, z0, z1))
ACTIVE(U22(tt, z0, z1)) → c3(PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(plus(z0, 0)) → c4
ACTIVE(plus(z0, s(z1))) → c5(U11'(tt, z1, z0))
ACTIVE(x(z0, 0)) → c6
ACTIVE(x(z0, s(z1))) → c7(U21'(tt, z1, z0))
ACTIVE(U11(z0, z1, z2)) → c8(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c9(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(tt) → c34
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c41
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:
active, U11, U12, s, plus, U21, U22, x, proper, top
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, 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
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 8 trailing nodes:
ACTIVE(U11(tt, z0, z1)) → c(U12'(tt, z0, z1))
PROPER(0) → c41
ACTIVE(plus(z0, s(z1))) → c5(U11'(tt, z1, z0))
PROPER(tt) → c34
ACTIVE(U21(tt, z0, z1)) → c2(U22'(tt, z0, z1))
ACTIVE(plus(z0, 0)) → c4
ACTIVE(x(z0, 0)) → c6
ACTIVE(x(z0, s(z1))) → c7(U21'(tt, z1, z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(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))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(U12(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c3(PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(U11(z0, z1, z2)) → c8(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c9(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
S tuples:
ACTIVE(U12(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c3(PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(U11(z0, z1, z2)) → c8(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c9(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:
active, U11, U12, s, plus, U21, U22, x, proper, top
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c1, c3, 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, c35, c36, c37, c38, c39, c40, c42, c43
(5) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(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))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(U11(z0, z1, z2)) → c8(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c9(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
S tuples:
ACTIVE(U11(z0, z1, z2)) → c8(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c9(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
K tuples:none
Defined Rule Symbols:
active, U11, U12, s, plus, U21, U22, x, proper, top
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
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, c35, c36, c37, c38, c39, c40, c42, c43, c
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(U11(z0, z1, z2)) → c8(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c9(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
S tuples:
ACTIVE(U11(z0, z1, z2)) → c8(U11'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U12(z0, z1, z2)) → c9(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
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, c35, c36, c37, c38, c39, c40, c42, c43, c
(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
U11(
z0,
z1,
z2)) →
c8(
U11'(
active(
z0),
z1,
z2),
ACTIVE(
z0)) by
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c8(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(U12(z0, z1, z2)) → c9(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c8(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
S tuples:
ACTIVE(U12(z0, z1, z2)) → c9(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c8(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
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, c35, c36, c37, c38, c39, c40, c42, c43, c, c8
(11) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(U12(z0, z1, z2)) → c9(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c8(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
S tuples:
ACTIVE(U12(z0, z1, z2)) → c9(U12'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c8(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
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, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8
(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
U12(
z0,
z1,
z2)) →
c9(
U12'(
active(
z0),
z1,
z2),
ACTIVE(
z0)) by
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c9(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c8(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c9(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
S tuples:
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c8(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c9(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9
(15) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c8(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c9(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
S tuples:
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c8(U11'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c9(U12'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9
(17) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(18) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
S tuples:
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1
(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
s(
z0)) →
c10(
S(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
(20) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))), ACTIVE(U11(tt, z0, z1)))
ACTIVE(s(U12(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10
(21) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(22) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U12(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U12(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10
(23) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(24) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(z0, z1, z2)) → c13(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2
(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
U21(
z0,
z1,
z2)) →
c13(
U21'(
active(
z0),
z1,
z2),
ACTIVE(
z0)) by
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c13(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c13(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
(26) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c13(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c13(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c13(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c13(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13
(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing tuple parts
(28) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c13(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c13(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c13(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c13(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13
(29) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(30) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c13(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U22(z0, z1, z2)) → c14(U22'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c13(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3
(31) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
U22(
z0,
z1,
z2)) →
c14(
U22'(
active(
z0),
z1,
z2),
ACTIVE(
z0)) by
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c14(U22'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c14(U22'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
(32) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c13(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c14(U22'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c14(U22'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c13(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2), ACTIVE(U11(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c14(U22'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c14(U22'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14
(33) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(34) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c13(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c14(U22'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c14(U22'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c8(U11'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c9(U12'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c10(S(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c13(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c14(U22'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c14(U22'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14
(35) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(36) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U11(z0, z1, z2)) → c33(U11'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4
(37) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
U11(
z0,
z1,
z2)) →
c33(
U11'(
proper(
z0),
proper(
z1),
proper(
z2)),
PROPER(
z0),
PROPER(
z1),
PROPER(
z2)) by
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
(38) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33
(39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing tuple parts
(40) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U12(z0, z1, z2)) → c35(U12'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c35, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33
(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
U12(
z0,
z1,
z2)) →
c35(
U12'(
proper(
z0),
proper(
z1),
proper(
z2)),
PROPER(
z0),
PROPER(
z1),
PROPER(
z2)) by
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
(42) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35
(43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing tuple parts
(44) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(s(z0)) → c36(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35
(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
s(
z0)) →
c36(
S(
proper(
z0)),
PROPER(
z0)) by
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(tt)) → c36(S(ok(tt)), PROPER(tt))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(0)) → c36(S(ok(0)), PROPER(0))
(46) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(tt)) → c36(S(ok(tt)), PROPER(tt))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(0)) → c36(S(ok(0)), PROPER(0))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(tt)) → c36(S(ok(tt)), PROPER(tt))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(0)) → c36(S(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35, c36
(47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(48) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(plus(z0, z1)) → c37(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c37, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35, c36, c36
(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
plus(
z0,
z1)) →
c37(
PLUS(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
(50) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35, c36, c36, c37
(51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(52) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U21(z0, z1, z2)) → c38(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c38, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35, c36, c36, c37, c37
(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
U21(
z0,
z1,
z2)) →
c38(
U21'(
proper(
z0),
proper(
z1),
proper(
z2)),
PROPER(
z0),
PROPER(
z1),
PROPER(
z2)) by
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
(54) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35, c36, c36, c37, c37, c38
(55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing tuple parts
(56) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(U22(z0, z1, z2)) → c39(U22'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c39, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35, c36, c36, c37, c37, c38, c38
(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
U22(
z0,
z1,
z2)) →
c39(
U22'(
proper(
z0),
proper(
z1),
proper(
z2)),
PROPER(
z0),
PROPER(
z1),
PROPER(
z2)) by
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
(58) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35, c36, c36, c37, c37, c38, c38, c39
(59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing tuple parts
(60) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PROPER(x(z0, z1)) → c40(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c40, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35, c36, c36, c37, c37, c38, c38, c39, c39
(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
x(
z0,
z1)) →
c40(
X(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(x(x0, U11(z0, z1, z2))) → c40(X(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(x(x0, tt)) → c40(X(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(x(x0, U12(z0, z1, z2))) → c40(X(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c40(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c40(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c40(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, U22(z0, z1, z2))) → c40(X(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c40(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, 0)) → c40(X(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(x(U11(z0, z1, z2), x1)) → c40(X(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(x(tt, x1)) → c40(X(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(x(U12(z0, z1, z2), x1)) → c40(X(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c40(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c40(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c40(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(U22(z0, z1, z2), x1)) → c40(X(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c40(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(0, x1)) → c40(X(ok(0), proper(x1)), PROPER(0), PROPER(x1))
(62) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1, z2))) → c40(X(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(x(x0, tt)) → c40(X(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(x(x0, U12(z0, z1, z2))) → c40(X(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c40(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c40(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c40(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, U22(z0, z1, z2))) → c40(X(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c40(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, 0)) → c40(X(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(x(U11(z0, z1, z2), x1)) → c40(X(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(x(tt, x1)) → c40(X(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(x(U12(z0, z1, z2), x1)) → c40(X(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c40(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c40(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c40(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(U22(z0, z1, z2), x1)) → c40(X(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c40(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(0, x1)) → c40(X(ok(0), proper(x1)), PROPER(0), PROPER(x1))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1, z2))) → c40(X(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(x(x0, tt)) → c40(X(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(x(x0, U12(z0, z1, z2))) → c40(X(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c40(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c40(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c40(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, U22(z0, z1, z2))) → c40(X(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c40(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, 0)) → c40(X(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(x(U11(z0, z1, z2), x1)) → c40(X(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(x(tt, x1)) → c40(X(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(x(U12(z0, z1, z2), x1)) → c40(X(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c40(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c40(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c40(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(U22(z0, z1, z2), x1)) → c40(X(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c40(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(0, x1)) → c40(X(ok(0), proper(x1)), PROPER(0), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, TOP, PROPER
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35, c36, c36, c37, c37, c38, c38, c39, c39, c40
(63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(64) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1, z2))) → c40(X(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(x(x0, U12(z0, z1, z2))) → c40(X(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c40(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c40(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c40(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, U22(z0, z1, z2))) → c40(X(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c40(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(U11(z0, z1, z2), x1)) → c40(X(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(x(U12(z0, z1, z2), x1)) → c40(X(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c40(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c40(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c40(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(U22(z0, z1, z2), x1)) → c40(X(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c40(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c40(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c40(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c40(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c40(X(ok(0), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
TOP(mark(z0)) → c42(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1, z2))) → c40(X(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(x(x0, U12(z0, z1, z2))) → c40(X(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c40(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c40(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c40(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, U22(z0, z1, z2))) → c40(X(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c40(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(U11(z0, z1, z2), x1)) → c40(X(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(x(U12(z0, z1, z2), x1)) → c40(X(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c40(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c40(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c40(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(U22(z0, z1, z2), x1)) → c40(X(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c40(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c40(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c40(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c40(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c40(X(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, TOP, PROPER
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c42, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35, c36, c36, c37, c37, c38, c38, c39, c39, c40, c40
(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
TOP(
mark(
z0)) →
c42(
TOP(
proper(
z0)),
PROPER(
z0)) by
TOP(mark(U11(z0, z1, z2))) → c42(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(tt)) → c42(TOP(ok(tt)), PROPER(tt))
TOP(mark(U12(z0, z1, z2))) → c42(TOP(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
TOP(mark(s(z0))) → c42(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c42(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c42(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(U22(z0, z1, z2))) → c42(TOP(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c42(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(0)) → c42(TOP(ok(0)), PROPER(0))
(66) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1, z2))) → c40(X(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(x(x0, U12(z0, z1, z2))) → c40(X(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c40(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c40(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c40(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, U22(z0, z1, z2))) → c40(X(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c40(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(U11(z0, z1, z2), x1)) → c40(X(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(x(U12(z0, z1, z2), x1)) → c40(X(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c40(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c40(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c40(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(U22(z0, z1, z2), x1)) → c40(X(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c40(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c40(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c40(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c40(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c40(X(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1, z2))) → c42(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(tt)) → c42(TOP(ok(tt)), PROPER(tt))
TOP(mark(U12(z0, z1, z2))) → c42(TOP(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
TOP(mark(s(z0))) → c42(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c42(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c42(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(U22(z0, z1, z2))) → c42(TOP(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c42(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(0)) → c42(TOP(ok(0)), PROPER(0))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1, z2))) → c40(X(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(x(x0, U12(z0, z1, z2))) → c40(X(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c40(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c40(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c40(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, U22(z0, z1, z2))) → c40(X(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c40(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(U11(z0, z1, z2), x1)) → c40(X(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(x(U12(z0, z1, z2), x1)) → c40(X(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c40(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c40(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c40(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(U22(z0, z1, z2), x1)) → c40(X(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c40(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c40(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c40(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c40(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c40(X(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1, z2))) → c42(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(tt)) → c42(TOP(ok(tt)), PROPER(tt))
TOP(mark(U12(z0, z1, z2))) → c42(TOP(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
TOP(mark(s(z0))) → c42(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c42(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c42(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(U22(z0, z1, z2))) → c42(TOP(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c42(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(0)) → c42(TOP(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, TOP, PROPER
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35, c36, c36, c37, c37, c38, c38, c39, c39, c40, c40, c42
(67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(68) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1, z2))) → c40(X(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(x(x0, U12(z0, z1, z2))) → c40(X(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c40(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c40(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c40(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, U22(z0, z1, z2))) → c40(X(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c40(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(U11(z0, z1, z2), x1)) → c40(X(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(x(U12(z0, z1, z2), x1)) → c40(X(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c40(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c40(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c40(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(U22(z0, z1, z2), x1)) → c40(X(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c40(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c40(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c40(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c40(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c40(X(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1, z2))) → c42(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(U12(z0, z1, z2))) → c42(TOP(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
TOP(mark(s(z0))) → c42(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c42(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c42(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(U22(z0, z1, z2))) → c42(TOP(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c42(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(tt)) → c42(TOP(ok(tt)))
TOP(mark(0)) → c42(TOP(ok(0)))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
TOP(ok(z0)) → c43(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1, z2))) → c40(X(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(x(x0, U12(z0, z1, z2))) → c40(X(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c40(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c40(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c40(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, U22(z0, z1, z2))) → c40(X(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c40(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(U11(z0, z1, z2), x1)) → c40(X(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(x(U12(z0, z1, z2), x1)) → c40(X(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c40(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c40(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c40(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(U22(z0, z1, z2), x1)) → c40(X(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c40(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c40(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c40(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c40(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c40(X(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1, z2))) → c42(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(U12(z0, z1, z2))) → c42(TOP(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
TOP(mark(s(z0))) → c42(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c42(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c42(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(U22(z0, z1, z2))) → c42(TOP(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c42(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(tt)) → c42(TOP(ok(tt)))
TOP(mark(0)) → c42(TOP(ok(0)))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, TOP, PROPER
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c43, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35, c36, c36, c37, c37, c38, c38, c39, c39, c40, c40, c42, c42
(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
TOP(
ok(
z0)) →
c43(
TOP(
active(
z0)),
ACTIVE(
z0)) by
TOP(ok(U11(tt, z0, z1))) → c43(TOP(mark(U12(tt, z0, z1))), ACTIVE(U11(tt, z0, z1)))
TOP(ok(U12(tt, z0, z1))) → c43(TOP(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
TOP(ok(U21(tt, z0, z1))) → c43(TOP(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(U22(tt, z0, z1))) → c43(TOP(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
TOP(ok(plus(z0, 0))) → c43(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c43(TOP(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c43(TOP(mark(0)), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c43(TOP(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(U11(z0, z1, z2))) → c43(TOP(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
TOP(ok(U12(z0, z1, z2))) → c43(TOP(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
TOP(ok(s(z0))) → c43(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c43(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c43(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c43(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(U22(z0, z1, z2))) → c43(TOP(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
TOP(ok(x(z0, z1))) → c43(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c43(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))
(70) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1, z2))) → c40(X(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(x(x0, U12(z0, z1, z2))) → c40(X(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c40(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c40(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c40(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, U22(z0, z1, z2))) → c40(X(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c40(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(U11(z0, z1, z2), x1)) → c40(X(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(x(U12(z0, z1, z2), x1)) → c40(X(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c40(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c40(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c40(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(U22(z0, z1, z2), x1)) → c40(X(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c40(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c40(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c40(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c40(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c40(X(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1, z2))) → c42(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(U12(z0, z1, z2))) → c42(TOP(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
TOP(mark(s(z0))) → c42(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c42(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c42(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(U22(z0, z1, z2))) → c42(TOP(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c42(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(tt)) → c42(TOP(ok(tt)))
TOP(mark(0)) → c42(TOP(ok(0)))
TOP(ok(U11(tt, z0, z1))) → c43(TOP(mark(U12(tt, z0, z1))), ACTIVE(U11(tt, z0, z1)))
TOP(ok(U12(tt, z0, z1))) → c43(TOP(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
TOP(ok(U21(tt, z0, z1))) → c43(TOP(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(U22(tt, z0, z1))) → c43(TOP(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
TOP(ok(plus(z0, 0))) → c43(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c43(TOP(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c43(TOP(mark(0)), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c43(TOP(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(U11(z0, z1, z2))) → c43(TOP(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
TOP(ok(U12(z0, z1, z2))) → c43(TOP(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
TOP(ok(s(z0))) → c43(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c43(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c43(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c43(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(U22(z0, z1, z2))) → c43(TOP(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
TOP(ok(x(z0, z1))) → c43(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c43(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1, z2))) → c40(X(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(x(x0, U12(z0, z1, z2))) → c40(X(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c40(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c40(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c40(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, U22(z0, z1, z2))) → c40(X(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c40(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(U11(z0, z1, z2), x1)) → c40(X(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(x(U12(z0, z1, z2), x1)) → c40(X(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c40(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c40(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c40(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(U22(z0, z1, z2), x1)) → c40(X(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c40(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c40(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c40(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c40(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c40(X(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1, z2))) → c42(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(U12(z0, z1, z2))) → c42(TOP(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
TOP(mark(s(z0))) → c42(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c42(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c42(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(U22(z0, z1, z2))) → c42(TOP(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c42(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(tt)) → c42(TOP(ok(tt)))
TOP(mark(0)) → c42(TOP(ok(0)))
TOP(ok(U11(tt, z0, z1))) → c43(TOP(mark(U12(tt, z0, z1))), ACTIVE(U11(tt, z0, z1)))
TOP(ok(U12(tt, z0, z1))) → c43(TOP(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
TOP(ok(U21(tt, z0, z1))) → c43(TOP(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(U22(tt, z0, z1))) → c43(TOP(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
TOP(ok(plus(z0, 0))) → c43(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c43(TOP(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c43(TOP(mark(0)), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c43(TOP(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(U11(z0, z1, z2))) → c43(TOP(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
TOP(ok(U12(z0, z1, z2))) → c43(TOP(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
TOP(ok(s(z0))) → c43(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c43(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c43(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c43(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(U22(z0, z1, z2))) → c43(TOP(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
TOP(ok(x(z0, z1))) → c43(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c43(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35, c36, c36, c37, c37, c38, c38, c39, c39, c40, c40, c42, c42, c43
(71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
TOP(mark(0)) → c42(TOP(ok(0)))
TOP(mark(tt)) → c42(TOP(ok(tt)))
(72) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1, z2))) → c40(X(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(x(x0, U12(z0, z1, z2))) → c40(X(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c40(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c40(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c40(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, U22(z0, z1, z2))) → c40(X(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c40(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(U11(z0, z1, z2), x1)) → c40(X(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(x(U12(z0, z1, z2), x1)) → c40(X(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c40(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c40(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c40(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(U22(z0, z1, z2), x1)) → c40(X(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c40(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c40(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c40(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c40(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c40(X(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1, z2))) → c42(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(U12(z0, z1, z2))) → c42(TOP(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
TOP(mark(s(z0))) → c42(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c42(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c42(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(U22(z0, z1, z2))) → c42(TOP(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c42(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(ok(U11(tt, z0, z1))) → c43(TOP(mark(U12(tt, z0, z1))), ACTIVE(U11(tt, z0, z1)))
TOP(ok(U12(tt, z0, z1))) → c43(TOP(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
TOP(ok(U21(tt, z0, z1))) → c43(TOP(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(U22(tt, z0, z1))) → c43(TOP(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
TOP(ok(plus(z0, 0))) → c43(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c43(TOP(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c43(TOP(mark(0)), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c43(TOP(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(U11(z0, z1, z2))) → c43(TOP(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
TOP(ok(U12(z0, z1, z2))) → c43(TOP(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
TOP(ok(s(z0))) → c43(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c43(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c43(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c43(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(U22(z0, z1, z2))) → c43(TOP(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
TOP(ok(x(z0, z1))) → c43(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c43(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))
S tuples:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1, z2))) → c40(X(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(x(x0, U12(z0, z1, z2))) → c40(X(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c40(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c40(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c40(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, U22(z0, z1, z2))) → c40(X(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c40(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(U11(z0, z1, z2), x1)) → c40(X(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(x(U12(z0, z1, z2), x1)) → c40(X(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c40(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c40(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c40(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(U22(z0, z1, z2), x1)) → c40(X(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c40(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c40(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c40(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c40(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c40(X(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1, z2))) → c42(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(U12(z0, z1, z2))) → c42(TOP(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
TOP(mark(s(z0))) → c42(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c42(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c42(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(U22(z0, z1, z2))) → c42(TOP(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c42(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(ok(U11(tt, z0, z1))) → c43(TOP(mark(U12(tt, z0, z1))), ACTIVE(U11(tt, z0, z1)))
TOP(ok(U12(tt, z0, z1))) → c43(TOP(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
TOP(ok(U21(tt, z0, z1))) → c43(TOP(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(U22(tt, z0, z1))) → c43(TOP(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
TOP(ok(plus(z0, 0))) → c43(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c43(TOP(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c43(TOP(mark(0)), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c43(TOP(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(U11(z0, z1, z2))) → c43(TOP(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
TOP(ok(U12(z0, z1, z2))) → c43(TOP(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
TOP(ok(s(z0))) → c43(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c43(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c43(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c43(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(U22(z0, z1, z2))) → c43(TOP(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
TOP(ok(x(z0, z1))) → c43(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c43(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
ACTIVE, U11', U12', S, PLUS, U21', U22', X, PROPER, TOP
Compound Symbols:
c11, c12, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c, c8, c8, c9, c9, c1, c10, c10, c2, c13, c13, c3, c14, c14, c4, c33, c33, c35, c35, c36, c36, c37, c37, c38, c38, c39, c39, c40, c40, c42, c43
(73) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c15(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c16(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(U12(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U12(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U22(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U22(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(U11(plus(z0, 0), x1, x2)) → c8(U11'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1, x2)) → c8(U11'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1, x2)) → c8(U11'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1, x2)) → c8(U11'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1, z2), x1, x2)) → c8(U11'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U11(U12(z0, z1, z2), x1, x2)) → c8(U11'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U11(s(z0), x1, x2)) → c8(U11'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1, x2)) → c8(U11'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1, x2)) → c8(U11'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(U22(z0, z1, z2), x1, x2)) → c8(U11'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1, x2)) → c8(U11'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U11(tt, z0, z1), x1, x2)) → c8(U11'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U12(plus(z0, 0), x1, x2)) → c9(U12'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)), x1, x2)) → c9(U12'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(x(z0, 0), x1, x2)) → c9(U12'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U12(x(z0, s(z1)), x1, x2)) → c9(U12'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U12(U11(z0, z1, z2), x1, x2)) → c9(U12'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U12(U12(z0, z1, z2), x1, x2)) → c9(U12'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U12(s(z0), x1, x2)) → c9(U12'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1), x1, x2)) → c9(U12'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U12(U21(z0, z1, z2), x1, x2)) → c9(U12'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U12(U22(z0, z1, z2), x1, x2)) → c9(U12'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(x(z0, z1), x1, x2)) → c9(U12'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U12(U11(tt, z0, z1), x1, x2)) → c9(U12'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(U11'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U11(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(U12'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U12(U12(tt, z0, z1), x1, x2)) → c1(ACTIVE(U12(tt, z0, z1)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c10(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c10(S(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1, z2))) → c10(S(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
ACTIVE(s(U12(z0, z1, z2))) → c10(S(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(U22(z0, z1, z2))) → c10(S(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c10(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c10(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(U11(tt, z0, z1))) → c10(S(mark(U12(tt, z0, z1))))
ACTIVE(s(U12(tt, z0, z1))) → c2(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U12(tt, z0, z1))) → c2(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c13(U21'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c13(U21'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c13(U21'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c13(U21'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1, z2), x1, x2)) → c13(U21'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U21(U12(z0, z1, z2), x1, x2)) → c13(U21'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c13(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c13(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c13(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(U22(z0, z1, z2), x1, x2)) → c13(U21'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c13(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U11(U21(tt, z0, z1), x1, x2)) → c8(U11'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U12(U21(tt, z0, z1), x1, x2)) → c9(U12'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(U22(tt, z0, z1))))
ACTIVE(U21(U11(tt, z0, z1), x1, x2)) → c13(U21'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c13(U21'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U12(tt, z0, z1), x1, x2)) → c3(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(plus(z0, 0), x1, x2)) → c14(U22'(mark(z0), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U22(plus(z0, s(z1)), x1, x2)) → c14(U22'(mark(U11(tt, z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U22(x(z0, 0), x1, x2)) → c14(U22'(mark(0), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U22(x(z0, s(z1)), x1, x2)) → c14(U22'(mark(U21(tt, z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U22(U11(z0, z1, z2), x1, x2)) → c14(U22'(U11(active(z0), z1, z2), x1, x2), ACTIVE(U11(z0, z1, z2)))
ACTIVE(U22(U12(z0, z1, z2), x1, x2)) → c14(U22'(U12(active(z0), z1, z2), x1, x2), ACTIVE(U12(z0, z1, z2)))
ACTIVE(U22(s(z0), x1, x2)) → c14(U22'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(plus(z0, z1), x1, x2)) → c14(U22'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U22(U21(z0, z1, z2), x1, x2)) → c14(U22'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U22(U22(z0, z1, z2), x1, x2)) → c14(U22'(U22(active(z0), z1, z2), x1, x2), ACTIVE(U22(z0, z1, z2)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(x(z0, z1), x1, x2)) → c14(U22'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U22(U11(tt, z0, z1), x1, x2)) → c14(U22'(mark(U12(tt, z0, z1)), x1, x2))
ACTIVE(U22(U21(tt, z0, z1), x1, x2)) → c14(U22'(mark(U22(tt, z0, z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(U11'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U11(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(U12'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U12(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(s(U22(tt, z0, z1))) → c4(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U22(tt, z0, z1))) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(U22'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U22(U12(tt, z0, z1), x1, x2)) → c4(ACTIVE(U12(tt, z0, z1)))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(U22'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U22(U22(tt, z0, z1), x1, x2)) → c4(ACTIVE(U22(tt, z0, z1)))
PROPER(U11(x0, x1, U11(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U11(x0, x1, U12(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U11(x0, x1, s(z0))) → c33(U11'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U11(x0, x1, plus(z0, z1))) → c33(U11'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U11(x0, x1, U21(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, x1, U22(z0, z1, z2))) → c33(U11'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U11(x0, x1, x(z0, z1))) → c33(U11'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U11(x0, U11(z0, z1, z2), x2)) → c33(U11'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U12(z0, z1, z2), x2)) → c33(U11'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, s(z0), x2)) → c33(U11'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U11(x0, plus(z0, z1), x2)) → c33(U11'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U11(x0, U21(z0, z1, z2), x2)) → c33(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, U22(z0, z1, z2), x2)) → c33(U11'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U11(x0, x(z0, z1), x2)) → c33(U11'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U11(U11(z0, z1, z2), x1, x2)) → c33(U11'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U12(z0, z1, z2), x1, x2)) → c33(U11'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(s(z0), x1, x2)) → c33(U11'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U11(plus(z0, z1), x1, x2)) → c33(U11'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(U21(z0, z1, z2), x1, x2)) → c33(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(U22(z0, z1, z2), x1, x2)) → c33(U11'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U11(x(z0, z1), x1, x2)) → c33(U11'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U11(x0, x1, tt)) → c33(U11'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, x1, 0)) → c33(U11'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U11(x0, tt, x2)) → c33(U11'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(x0, 0, x2)) → c33(U11'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U11(tt, x1, x2)) → c33(U11'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U11(0, x1, x2)) → c33(U11'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, U11(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U12(x0, x1, U12(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U12(x0, x1, s(z0))) → c35(U12'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U12(x0, x1, plus(z0, z1))) → c35(U12'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U12(x0, x1, U21(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U12(x0, x1, U22(z0, z1, z2))) → c35(U12'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U12(x0, x1, x(z0, z1))) → c35(U12'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U12(x0, U11(z0, z1, z2), x2)) → c35(U12'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U12(z0, z1, z2), x2)) → c35(U12'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, s(z0), x2)) → c35(U12'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U12(x0, plus(z0, z1), x2)) → c35(U12'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U12(x0, U21(z0, z1, z2), x2)) → c35(U12'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, U22(z0, z1, z2), x2)) → c35(U12'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U12(x0, x(z0, z1), x2)) → c35(U12'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U12(U11(z0, z1, z2), x1, x2)) → c35(U12'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U12(z0, z1, z2), x1, x2)) → c35(U12'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(s(z0), x1, x2)) → c35(U12'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U12(plus(z0, z1), x1, x2)) → c35(U12'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(U21(z0, z1, z2), x1, x2)) → c35(U12'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(U22(z0, z1, z2), x1, x2)) → c35(U12'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U12(x(z0, z1), x1, x2)) → c35(U12'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U12(x0, x1, tt)) → c35(U12'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, x1, 0)) → c35(U12'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U12(x0, tt, x2)) → c35(U12'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(x0, 0, x2)) → c35(U12'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U12(tt, x1, x2)) → c35(U12'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U12(0, x1, x2)) → c35(U12'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1, z2))) → c36(S(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
PROPER(s(U12(z0, z1, z2))) → c36(S(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
PROPER(s(s(z0))) → c36(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c36(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c36(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(U22(z0, z1, z2))) → c36(S(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c36(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c36(S(ok(tt)))
PROPER(s(0)) → c36(S(ok(0)))
PROPER(plus(x0, U11(z0, z1, z2))) → c37(PLUS(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(plus(x0, U12(z0, z1, z2))) → c37(PLUS(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c37(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c37(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c37(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, U22(z0, z1, z2))) → c37(PLUS(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c37(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(U11(z0, z1, z2), x1)) → c37(PLUS(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(plus(U12(z0, z1, z2), x1)) → c37(PLUS(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c37(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c37(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c37(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(U22(z0, z1, z2), x1)) → c37(PLUS(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c37(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c37(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c37(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c37(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c37(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U21(x0, x1, U12(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c38(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c38(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, U22(z0, z1, z2))) → c38(U21'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c38(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, U11(z0, z1, z2), x2)) → c38(U21'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U12(z0, z1, z2), x2)) → c38(U21'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c38(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c38(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c38(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, U22(z0, z1, z2), x2)) → c38(U21'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c38(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(U11(z0, z1, z2), x1, x2)) → c38(U21'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U12(z0, z1, z2), x1, x2)) → c38(U21'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c38(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c38(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c38(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(U22(z0, z1, z2), x1, x2)) → c38(U21'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c38(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c38(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c38(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c38(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c38(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c38(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c38(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, U11(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1, z2)))
PROPER(U22(x0, x1, U12(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U12(z0, z1, z2)))
PROPER(U22(x0, x1, s(z0))) → c39(U22'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U22(x0, x1, plus(z0, z1))) → c39(U22'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U22(x0, x1, U21(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U22(x0, x1, U22(z0, z1, z2))) → c39(U22'(proper(x0), proper(x1), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U22(z0, z1, z2)))
PROPER(U22(x0, x1, x(z0, z1))) → c39(U22'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U22(x0, U11(z0, z1, z2), x2)) → c39(U22'(proper(x0), U11(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U12(z0, z1, z2), x2)) → c39(U22'(proper(x0), U12(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U12(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, s(z0), x2)) → c39(U22'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U22(x0, plus(z0, z1), x2)) → c39(U22'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U22(x0, U21(z0, z1, z2), x2)) → c39(U22'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, U22(z0, z1, z2), x2)) → c39(U22'(proper(x0), U22(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U22(z0, z1, z2)), PROPER(x2))
PROPER(U22(x0, x(z0, z1), x2)) → c39(U22'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U22(U11(z0, z1, z2), x1, x2)) → c39(U22'(U11(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U11(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U12(z0, z1, z2), x1, x2)) → c39(U22'(U12(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U12(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(s(z0), x1, x2)) → c39(U22'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U22(plus(z0, z1), x1, x2)) → c39(U22'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(U21(z0, z1, z2), x1, x2)) → c39(U22'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(U22(z0, z1, z2), x1, x2)) → c39(U22'(U22(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U22(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U22(x(z0, z1), x1, x2)) → c39(U22'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U22(x0, x1, tt)) → c39(U22'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, x1, 0)) → c39(U22'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U22(x0, tt, x2)) → c39(U22'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(x0, 0, x2)) → c39(U22'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U22(tt, x1, x2)) → c39(U22'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U22(0, x1, x2)) → c39(U22'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1, z2))) → c40(X(proper(x0), U11(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U11(z0, z1, z2)))
PROPER(x(x0, U12(z0, z1, z2))) → c40(X(proper(x0), U12(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U12(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c40(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c40(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c40(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, U22(z0, z1, z2))) → c40(X(proper(x0), U22(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U22(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c40(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(U11(z0, z1, z2), x1)) → c40(X(U11(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U11(z0, z1, z2)), PROPER(x1))
PROPER(x(U12(z0, z1, z2), x1)) → c40(X(U12(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U12(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c40(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c40(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c40(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(U22(z0, z1, z2), x1)) → c40(X(U22(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U22(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c40(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c40(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c40(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c40(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c40(X(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1, z2))) → c42(TOP(U11(proper(z0), proper(z1), proper(z2))), PROPER(U11(z0, z1, z2)))
TOP(mark(U12(z0, z1, z2))) → c42(TOP(U12(proper(z0), proper(z1), proper(z2))), PROPER(U12(z0, z1, z2)))
TOP(mark(s(z0))) → c42(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c42(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c42(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(U22(z0, z1, z2))) → c42(TOP(U22(proper(z0), proper(z1), proper(z2))), PROPER(U22(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c42(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(ok(U11(tt, z0, z1))) → c43(TOP(mark(U12(tt, z0, z1))), ACTIVE(U11(tt, z0, z1)))
TOP(ok(U12(tt, z0, z1))) → c43(TOP(mark(s(plus(z1, z0)))), ACTIVE(U12(tt, z0, z1)))
TOP(ok(U21(tt, z0, z1))) → c43(TOP(mark(U22(tt, z0, z1))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(U22(tt, z0, z1))) → c43(TOP(mark(plus(x(z1, z0), z1))), ACTIVE(U22(tt, z0, z1)))
TOP(ok(plus(z0, 0))) → c43(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c43(TOP(mark(U11(tt, z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c43(TOP(mark(0)), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c43(TOP(mark(U21(tt, z1, z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(U11(z0, z1, z2))) → c43(TOP(U11(active(z0), z1, z2)), ACTIVE(U11(z0, z1, z2)))
TOP(ok(U12(z0, z1, z2))) → c43(TOP(U12(active(z0), z1, z2)), ACTIVE(U12(z0, z1, z2)))
TOP(ok(s(z0))) → c43(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c43(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c43(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c43(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(U22(z0, z1, z2))) → c43(TOP(U22(active(z0), z1, z2)), ACTIVE(U22(z0, z1, z2)))
TOP(ok(x(z0, z1))) → c43(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c43(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))
(74) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
S tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
K tuples:none
Defined Rule Symbols:
active, s, plus, x, U11, U12, U21, U22, proper
Defined Pair Symbols:
U11', U12', S, PLUS, U21', U22', X
Compound Symbols:
c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32
(75) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
active(U11(z0, z1, z2)) → U11(active(z0), z1, z2)
active(U12(z0, z1, z2)) → U12(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(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(U22(z0, z1, z2)) → U22(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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, z2) → mark(U12(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U22(mark(z0), z1, z2) → mark(U22(z0, z1, z2))
U22(ok(z0), ok(z1), ok(z2)) → ok(U22(z0, z1, z2))
proper(U11(z0, z1, z2)) → U11(proper(z0), proper(z1), proper(z2))
proper(tt) → ok(tt)
proper(U12(z0, z1, z2)) → U12(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(U22(z0, z1, z2)) → U22(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(0) → ok(0)
(76) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
S tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', S, PLUS, U21', U22', X
Compound Symbols:
c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32
(77) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(PLUS(x1, x2)) = 0
POL(S(x1)) = 0
POL(U11'(x1, x2, x3)) = 0
POL(U12'(x1, x2, x3)) = [4]x2 + [4]x3
POL(U21'(x1, x2, x3)) = x2
POL(U22'(x1, x2, x3)) = [2]x3
POL(X(x1, x2)) = 0
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(mark(x1)) = [5]
POL(ok(x1)) = [1] + x1
(78) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
S tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
K tuples:
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', S, PLUS, U21', U22', X
Compound Symbols:
c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32
(79) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(PLUS(x1, x2)) = 0
POL(S(x1)) = 0
POL(U11'(x1, x2, x3)) = x2
POL(U12'(x1, x2, x3)) = [4]x2 + [4]x3
POL(U21'(x1, x2, x3)) = [3]x2
POL(U22'(x1, x2, x3)) = 0
POL(X(x1, x2)) = 0
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(mark(x1)) = 0
POL(ok(x1)) = [4] + x1
(80) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
S tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
K tuples:
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', S, PLUS, U21', U22', X
Compound Symbols:
c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32
(81) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
X(z0, mark(z1)) → c31(X(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(PLUS(x1, x2)) = 0
POL(S(x1)) = 0
POL(U11'(x1, x2, x3)) = [2]x2
POL(U12'(x1, x2, x3)) = x1
POL(U21'(x1, x2, x3)) = x2
POL(U22'(x1, x2, x3)) = 0
POL(X(x1, x2)) = x2
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(mark(x1)) = [1] + x1
POL(ok(x1)) = x1
(82) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
S tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
K tuples:
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
X(z0, mark(z1)) → c31(X(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', S, PLUS, U21', U22', X
Compound Symbols:
c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32
(83) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(PLUS(x1, x2)) = 0
POL(S(x1)) = [4]x1
POL(U11'(x1, x2, x3)) = [3]x2 + [5]x3
POL(U12'(x1, x2, x3)) = [2]x2 + [3]x3
POL(U21'(x1, x2, x3)) = [2]x1 + [5]x2 + [2]x3
POL(U22'(x1, x2, x3)) = x2 + [2]x3
POL(X(x1, x2)) = [2]x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(mark(x1)) = [1] + x1
POL(ok(x1)) = [4] + x1
(84) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
S tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
K tuples:
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
X(z0, mark(z1)) → c31(X(z0, z1))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', S, PLUS, U21', U22', X
Compound Symbols:
c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32
(85) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(PLUS(x1, x2)) = [4]x2
POL(S(x1)) = 0
POL(U11'(x1, x2, x3)) = [4]x2 + [4]x3
POL(U12'(x1, x2, x3)) = [2]x1 + [4]x2 + [4]x3
POL(U21'(x1, x2, x3)) = [2]x2 + [3]x3
POL(U22'(x1, x2, x3)) = 0
POL(X(x1, x2)) = [2]x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(mark(x1)) = [4] + x1
POL(ok(x1)) = [4] + x1
(86) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
S tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
K tuples:
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
X(z0, mark(z1)) → c31(X(z0, z1))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', S, PLUS, U21', U22', X
Compound Symbols:
c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32
(87) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(PLUS(x1, x2)) = x1
POL(S(x1)) = 0
POL(U11'(x1, x2, x3)) = [4]x2 + [2]x3
POL(U12'(x1, x2, x3)) = [5]x2
POL(U21'(x1, x2, x3)) = [2]x2 + [4]x3
POL(U22'(x1, x2, x3)) = [2]x2 + [4]x3
POL(X(x1, x2)) = 0
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(mark(x1)) = [1] + x1
POL(ok(x1)) = x1
(88) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
S tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
K tuples:
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
X(z0, mark(z1)) → c31(X(z0, z1))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', S, PLUS, U21', U22', X
Compound Symbols:
c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32
(89) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(PLUS(x1, x2)) = [4]x2
POL(S(x1)) = 0
POL(U11'(x1, x2, x3)) = [2]x2 + [3]x3
POL(U12'(x1, x2, x3)) = 0
POL(U21'(x1, x2, x3)) = [2]x1 + [2]x2 + [4]x3
POL(U22'(x1, x2, x3)) = [2]x1 + [3]x2
POL(X(x1, x2)) = [4]x1 + [4]x2
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(mark(x1)) = [4] + x1
POL(ok(x1)) = x1
(90) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
S tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
K tuples:
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
X(z0, mark(z1)) → c31(X(z0, z1))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', S, PLUS, U21', U22', X
Compound Symbols:
c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32
(91) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(PLUS(x1, x2)) = [2]x1
POL(S(x1)) = [4]x1
POL(U11'(x1, x2, x3)) = [2]x1 + [4]x3
POL(U12'(x1, x2, x3)) = [2]x1 + [4]x3
POL(U21'(x1, x2, x3)) = [2]x1 + [4]x2 + x3
POL(U22'(x1, x2, x3)) = [4]x1 + [2]x2 + [3]x3
POL(X(x1, x2)) = 0
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(mark(x1)) = [1] + x1
POL(ok(x1)) = x1
(92) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(z0, mark(z1)) → c31(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
S tuples:none
K tuples:
U12'(ok(z0), ok(z1), ok(z2)) → c20(U12'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c27(U21'(z0, z1, z2))
U22'(ok(z0), ok(z1), ok(z2)) → c29(U22'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c18(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c19(U12'(z0, z1, z2))
X(z0, mark(z1)) → c31(X(z0, z1))
S(mark(z0)) → c21(S(z0))
S(ok(z0)) → c22(S(z0))
U21'(mark(z0), z1, z2) → c26(U21'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c32(X(z0, z1))
PLUS(z0, mark(z1)) → c24(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c25(PLUS(z0, z1))
PLUS(mark(z0), z1) → c23(PLUS(z0, z1))
U22'(mark(z0), z1, z2) → c28(U22'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
Defined Rule Symbols:none
Defined Pair Symbols:
U11', U12', S, PLUS, U21', U22', X
Compound Symbols:
c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32
(93) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(94) BOUNDS(O(1), O(1))