(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(