Runtime Complexity (innermost) proof of /tmp/tmpNFDjvr/LengthOfFiniteLists_nosorts_noand

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 87 ms)

↳2 CdtProblem

    ↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳4 CdtProblem

    ↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳6 CdtProblem

    ↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 66 ms)

    ↳8 CdtProblem

    ↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳10 CdtProblem

    ↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 20 ms)

    ↳12 CdtProblem

    ↳13 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳14 CdtProblem

    ↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳16 CdtProblem

    ↳17 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳18 CdtProblem

    ↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳20 CdtProblem

    ↳21 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳22 CdtProblem

    ↳23 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳24 CdtProblem

    ↳25 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳26 CdtProblem

    ↳27 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳28 CdtProblem

    ↳29 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳30 CdtProblem

    ↳31 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳32 CdtProblem

    ↳33 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳34 CdtProblem

    ↳35 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳36 CdtProblem

    ↳37 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳38 CdtProblem

↳39 CpxTrsMatchBoundsTAProof (⇔, 268 ms)

↳40 BOUNDS(O(1), O(n^1))

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), S(length(z0)), LENGTH(z0))
ACTIVE(length(nil)) → c3(MARK(0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(cons(z0, z1)) → c6(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(0) → c7(ACTIVE(0))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(tt) → c9(ACTIVE(tt))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(nil) → c13(ACTIVE(nil))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))

S tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), S(length(z0)), LENGTH(z0))
ACTIVE(length(nil)) → c3(MARK(0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(cons(z0, z1)) → c6(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(0) → c7(ACTIVE(0))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(tt) → c9(ACTIVE(tt))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(nil) → c13(ACTIVE(nil))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

ACTIVE(length(nil)) → c3(MARK(0))
MARK(0) → c7(ACTIVE(0))
MARK(nil) → c13(ACTIVE(nil))
MARK(tt) → c9(ACTIVE(tt))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), S(length(z0)), LENGTH(z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(cons(z0, z1)) → c6(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))

S tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), S(length(z0)), LENGTH(z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(cons(z0, z1)) → c6(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c, c1, c2, c4, c5, c6, c8, c10, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(cons(z0, z1)) → c6(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(cons(z0, z1)) → c6(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c6, c8, c10, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2

(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace MARK(cons(z0, z1)) → c6(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0)) by

MARK(cons(z0, z1)) → c6(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(z0, z1)) → c6(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(z0, z1)) → c6(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c8, c10, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c8, c10, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6

(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace MARK(U11(z0, z1)) → c8(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0)) by

MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1), MARK(0))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1), MARK(tt))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1), MARK(nil))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1), MARK(0))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1), MARK(tt))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1), MARK(nil))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1), MARK(0))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1), MARK(tt))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1), MARK(nil))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c10, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6, c8, c8

(13) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c10, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6, c8, c8, c8

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace MARK(U12(z0, z1)) → c10(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0)) by

MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1), MARK(0))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1), MARK(tt))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1), MARK(nil))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1), MARK(0))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1), MARK(tt))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1), MARK(nil))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1), MARK(0))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1), MARK(tt))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1), MARK(nil))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6, c8, c8, c8, c10, c10

(17) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6, c8, c8, c8, c10, c10, c10

(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0)) by

MARK(s(z0)) → c11(ACTIVE(s(z0)), S(mark(z0)), MARK(z0))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)), MARK(0))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)), MARK(tt))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)), MARK(nil))
MARK(s(x0)) → c11

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(z0)) → c11(ACTIVE(s(z0)), S(mark(z0)), MARK(z0))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)), MARK(0))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)), MARK(tt))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)), MARK(nil))
MARK(s(x0)) → c11

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(z0)) → c11(ACTIVE(s(z0)), S(mark(z0)), MARK(z0))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)), MARK(0))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)), MARK(tt))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)), MARK(nil))
MARK(s(x0)) → c11

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6, c8, c8, c8, c10, c10, c10, c11, c11

(21) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

MARK(s(x0)) → c11

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(z0)) → c11(ACTIVE(s(z0)), S(mark(z0)), MARK(z0))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)), MARK(0))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)), MARK(tt))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)), MARK(nil))

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(z0)) → c11(ACTIVE(s(z0)), S(mark(z0)), MARK(z0))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)), MARK(0))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)), MARK(tt))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)), MARK(nil))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6, c8, c8, c8, c10, c10, c10, c11

(23) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6, c8, c8, c8, c10, c10, c10, c11, c11

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace MARK(length(z0)) → c12(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0)) by

MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(x0)) → c12

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(x0)) → c12

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(x0)) → c12

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6, c8, c8, c8, c10, c10, c10, c11, c11, c12, c12

(27) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

MARK(length(x0)) → c12

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6, c8, c8, c8, c10, c10, c10, c11, c11, c12

(29) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6, c8, c8, c8, c10, c10, c10, c11, c11, c12, c12

(31) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace MARK(cons(zeros, x1)) → c6(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros)) by

MARK(cons(zeros, z1)) → c6(ACTIVE(cons(zeros, z1)), CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(zeros, x0)) → c6(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c6

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(cons(zeros, z1)) → c6(ACTIVE(cons(zeros, z1)), CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(zeros, x0)) → c6(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c6

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(cons(zeros, z1)) → c6(ACTIVE(cons(zeros, z1)), CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(zeros, x0)) → c6(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c6

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6, c8, c8, c8, c10, c10, c10, c11, c11, c12, c12, c6

(33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

MARK(cons(zeros, x0)) → c6

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(cons(zeros, z1)) → c6(ACTIVE(cons(zeros, z1)), CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(zeros, x0)) → c6(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(cons(zeros, z1)) → c6(ACTIVE(cons(zeros, z1)), CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(zeros, x0)) → c6(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6, c8, c8, c8, c10, c10, c10, c11, c11, c12, c12

(35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(cons(zeros, x0)) → c6(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, z1)) → c6(CONS(mark(zeros), z1), MARK(zeros))

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(cons(zeros, x0)) → c6(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, z1)) → c6(CONS(mark(zeros), z1), MARK(zeros))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6, c8, c8, c8, c10, c10, c10, c11, c11, c12, c12

(37) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace MARK(cons(cons(z0, z1), x1)) → c6(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1))) by

MARK(cons(cons(x0, x1), z1)) → c6(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c6(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c6(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c6(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c6(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(U11(z0, z1), x1), x2)) → c6(ACTIVE(cons(active(cons(active(U11(mark(z0), z1)), x1)), x2)), CONS(mark(cons(U11(z0, z1), x1)), x2), MARK(cons(U11(z0, z1), x1)))
MARK(cons(cons(tt, x1), x2)) → c6(ACTIVE(cons(active(cons(active(tt), x1)), x2)), CONS(mark(cons(tt, x1)), x2), MARK(cons(tt, x1)))
MARK(cons(cons(U12(z0, z1), x1), x2)) → c6(ACTIVE(cons(active(cons(active(U12(mark(z0), z1)), x1)), x2)), CONS(mark(cons(U12(z0, z1), x1)), x2), MARK(cons(U12(z0, z1), x1)))
MARK(cons(cons(s(z0), x1), x2)) → c6(ACTIVE(cons(active(cons(active(s(mark(z0))), x1)), x2)), CONS(mark(cons(s(z0), x1)), x2), MARK(cons(s(z0), x1)))
MARK(cons(cons(length(z0), x1), x2)) → c6(ACTIVE(cons(active(cons(active(length(mark(z0))), x1)), x2)), CONS(mark(cons(length(z0), x1)), x2), MARK(cons(length(z0), x1)))
MARK(cons(cons(nil, x1), x2)) → c6(ACTIVE(cons(active(cons(active(nil), x1)), x2)), CONS(mark(cons(nil, x1)), x2), MARK(cons(nil, x1)))
MARK(cons(cons(x0, x1), x2)) → c6(CONS(mark(cons(x0, x1)), x2))

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, z0)) → mark(U12(tt, z0))
active(U12(tt, z0)) → mark(s(length(z0)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(U11(tt, z1))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)

Tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(cons(zeros, x0)) → c6(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, z1)) → c6(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c6(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c6(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c6(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c6(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c6(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(U11(z0, z1), x1), x2)) → c6(ACTIVE(cons(active(cons(active(U11(mark(z0), z1)), x1)), x2)), CONS(mark(cons(U11(z0, z1), x1)), x2), MARK(cons(U11(z0, z1), x1)))
MARK(cons(cons(tt, x1), x2)) → c6(ACTIVE(cons(active(cons(active(tt), x1)), x2)), CONS(mark(cons(tt, x1)), x2), MARK(cons(tt, x1)))
MARK(cons(cons(U12(z0, z1), x1), x2)) → c6(ACTIVE(cons(active(cons(active(U12(mark(z0), z1)), x1)), x2)), CONS(mark(cons(U12(z0, z1), x1)), x2), MARK(cons(U12(z0, z1), x1)))
MARK(cons(cons(s(z0), x1), x2)) → c6(ACTIVE(cons(active(cons(active(s(mark(z0))), x1)), x2)), CONS(mark(cons(s(z0), x1)), x2), MARK(cons(s(z0), x1)))
MARK(cons(cons(length(z0), x1), x2)) → c6(ACTIVE(cons(active(cons(active(length(mark(z0))), x1)), x2)), CONS(mark(cons(length(z0), x1)), x2), MARK(cons(length(z0), x1)))
MARK(cons(cons(nil, x1), x2)) → c6(ACTIVE(cons(active(cons(active(nil), x1)), x2)), CONS(mark(cons(nil, x1)), x2), MARK(cons(nil, x1)))
MARK(cons(cons(x0, x1), x2)) → c6(CONS(mark(cons(x0, x1)), x2))

S tuples:

ACTIVE(U11(tt, z0)) → c1(MARK(U12(tt, z0)), U12'(tt, z0))
ACTIVE(length(cons(z0, z1))) → c4(MARK(U11(tt, z1)), U11'(tt, z1))
MARK(zeros) → c5(ACTIVE(zeros))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
U11'(mark(z0), z1) → c18(U11'(z0, z1))
U11'(z0, mark(z1)) → c19(U11'(z0, z1))
U11'(active(z0), z1) → c20(U11'(z0, z1))
U11'(z0, active(z1)) → c21(U11'(z0, z1))
U12'(mark(z0), z1) → c22(U12'(z0, z1))
U12'(z0, mark(z1)) → c23(U12'(z0, z1))
U12'(active(z0), z1) → c24(U12'(z0, z1))
U12'(z0, active(z1)) → c25(U12'(z0, z1))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
LENGTH(mark(z0)) → c28(LENGTH(z0))
LENGTH(active(z0)) → c29(LENGTH(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(U12(tt, z0)) → c2(MARK(s(length(z0))), LENGTH(z0))
MARK(cons(U11(z0, z1), x1)) → c6(ACTIVE(cons(active(U11(mark(z0), z1)), x1)), CONS(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(cons(U12(z0, z1), x1)) → c6(ACTIVE(cons(active(U12(mark(z0), z1)), x1)), CONS(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(cons(s(z0), x1)) → c6(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(length(z0), x1)) → c6(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(x0, x1)) → c6(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c6(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c6(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c6(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c6(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(z0, z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U11(zeros, x1)) → c8(ACTIVE(U11(active(zeros), x1)), U11'(mark(zeros), x1), MARK(zeros))
MARK(U11(cons(z0, z1), x1)) → c8(ACTIVE(U11(active(cons(mark(z0), z1)), x1)), U11'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U11(U11(z0, z1), x1)) → c8(ACTIVE(U11(active(U11(mark(z0), z1)), x1)), U11'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U11(U12(z0, z1), x1)) → c8(ACTIVE(U11(active(U12(mark(z0), z1)), x1)), U11'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U11(s(z0), x1)) → c8(ACTIVE(U11(active(s(mark(z0))), x1)), U11'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U11(length(z0), x1)) → c8(ACTIVE(U11(active(length(mark(z0))), x1)), U11'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U11(x0, x1)) → c8(U11'(mark(x0), x1))
MARK(U11(0, x1)) → c8(ACTIVE(U11(active(0), x1)), U11'(mark(0), x1))
MARK(U11(tt, x1)) → c8(ACTIVE(U11(active(tt), x1)), U11'(mark(tt), x1))
MARK(U11(nil, x1)) → c8(ACTIVE(U11(active(nil), x1)), U11'(mark(nil), x1))
MARK(U12(z0, z1)) → c10(ACTIVE(U12(z0, z1)), U12'(mark(z0), z1), MARK(z0))
MARK(U12(zeros, x1)) → c10(ACTIVE(U12(active(zeros), x1)), U12'(mark(zeros), x1), MARK(zeros))
MARK(U12(cons(z0, z1), x1)) → c10(ACTIVE(U12(active(cons(mark(z0), z1)), x1)), U12'(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(U12(U11(z0, z1), x1)) → c10(ACTIVE(U12(active(U11(mark(z0), z1)), x1)), U12'(mark(U11(z0, z1)), x1), MARK(U11(z0, z1)))
MARK(U12(U12(z0, z1), x1)) → c10(ACTIVE(U12(active(U12(mark(z0), z1)), x1)), U12'(mark(U12(z0, z1)), x1), MARK(U12(z0, z1)))
MARK(U12(s(z0), x1)) → c10(ACTIVE(U12(active(s(mark(z0))), x1)), U12'(mark(s(z0)), x1), MARK(s(z0)))
MARK(U12(length(z0), x1)) → c10(ACTIVE(U12(active(length(mark(z0))), x1)), U12'(mark(length(z0)), x1), MARK(length(z0)))
MARK(U12(x0, x1)) → c10(U12'(mark(x0), x1))
MARK(U12(0, x1)) → c10(ACTIVE(U12(active(0), x1)), U12'(mark(0), x1))
MARK(U12(tt, x1)) → c10(ACTIVE(U12(active(tt), x1)), U12'(mark(tt), x1))
MARK(U12(nil, x1)) → c10(ACTIVE(U12(active(nil), x1)), U12'(mark(nil), x1))
MARK(s(zeros)) → c11(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c11(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(U11(z0, z1))) → c11(ACTIVE(s(active(U11(mark(z0), z1)))), S(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(s(U12(z0, z1))) → c11(ACTIVE(s(active(U12(mark(z0), z1)))), S(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(s(s(z0))) → c11(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(length(z0))) → c11(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(z0)) → c11(S(mark(z0)), MARK(z0))
MARK(s(0)) → c11(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c11(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c11(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(length(z0)) → c12(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c12(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c12(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(U11(z0, z1))) → c12(ACTIVE(length(active(U11(mark(z0), z1)))), LENGTH(mark(U11(z0, z1))), MARK(U11(z0, z1)))
MARK(length(U12(z0, z1))) → c12(ACTIVE(length(active(U12(mark(z0), z1)))), LENGTH(mark(U12(z0, z1))), MARK(U12(z0, z1)))
MARK(length(s(z0))) → c12(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(length(z0))) → c12(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(0)) → c12(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c12(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c12(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(cons(zeros, x0)) → c6(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, z1)) → c6(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c6(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c6(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c6(ACTIVE(cons(active(cons(active(zeros), x1)), x2)), CONS(mark(cons(zeros, x1)), x2), MARK(cons(zeros, x1)))
MARK(cons(cons(cons(z0, z1), x1), x2)) → c6(ACTIVE(cons(active(cons(active(cons(mark(z0), z1)), x1)), x2)), CONS(mark(cons(cons(z0, z1), x1)), x2), MARK(cons(cons(z0, z1), x1)))
MARK(cons(cons(0, x1), x2)) → c6(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(U11(z0, z1), x1), x2)) → c6(ACTIVE(cons(active(cons(active(U11(mark(z0), z1)), x1)), x2)), CONS(mark(cons(U11(z0, z1), x1)), x2), MARK(cons(U11(z0, z1), x1)))
MARK(cons(cons(tt, x1), x2)) → c6(ACTIVE(cons(active(cons(active(tt), x1)), x2)), CONS(mark(cons(tt, x1)), x2), MARK(cons(tt, x1)))
MARK(cons(cons(U12(z0, z1), x1), x2)) → c6(ACTIVE(cons(active(cons(active(U12(mark(z0), z1)), x1)), x2)), CONS(mark(cons(U12(z0, z1), x1)), x2), MARK(cons(U12(z0, z1), x1)))
MARK(cons(cons(s(z0), x1), x2)) → c6(ACTIVE(cons(active(cons(active(s(mark(z0))), x1)), x2)), CONS(mark(cons(s(z0), x1)), x2), MARK(cons(s(z0), x1)))
MARK(cons(cons(length(z0), x1), x2)) → c6(ACTIVE(cons(active(cons(active(length(mark(z0))), x1)), x2)), CONS(mark(cons(length(z0), x1)), x2), MARK(cons(length(z0), x1)))
MARK(cons(cons(nil, x1), x2)) → c6(ACTIVE(cons(active(cons(active(nil), x1)), x2)), CONS(mark(cons(nil, x1)), x2), MARK(cons(nil, x1)))
MARK(cons(cons(x0, x1), x2)) → c6(CONS(mark(cons(x0, x1)), x2))

K tuples:none
Defined Rule Symbols:

active, mark, cons, U11, U12, s, length

Defined Pair Symbols:

ACTIVE, MARK, CONS, U11', U12', S, LENGTH

Compound Symbols:

c1, c4, c5, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c, c2, c6, c6, c6, c8, c8, c8, c10, c10, c10, c11, c11, c12, c12

(39) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 4.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5, 6, 7]
transitions:
zeros0() → 0
00() → 0
tt0() → 0
nil0() → 0
active0(0) → 1
mark0(0) → 2
cons0(0, 0) → 3
U110(0, 0) → 4
U120(0, 0) → 5
s0(0) → 6
length0(0) → 7
01() → 9
zeros1() → 10
cons1(9, 10) → 8
mark1(8) → 1
zeros1() → 11
active1(11) → 2
01() → 12
active1(12) → 2
tt1() → 13
active1(13) → 2
nil1() → 14
active1(14) → 2
02() → 16
zeros2() → 17
cons2(16, 17) → 15
mark2(15) → 2
mark2(9) → 19
cons2(19, 10) → 18
active2(18) → 1
mark3(16) → 21
cons3(21, 17) → 20
active3(20) → 2
02() → 22
active2(22) → 19
cons3(9, 10) → 18
cons3(22, 10) → 18
03() → 23
active3(23) → 21
cons4(16, 17) → 20
cons4(23, 17) → 20

(0) Obligation:

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

(2) Obligation:

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(4) Obligation:

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(6) Obligation:

(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(8) Obligation:

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(10) Obligation:

(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(12) Obligation:

(13) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(14) Obligation:

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(16) Obligation:

(17) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(18) Obligation:

(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(20) Obligation:

(21) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(22) Obligation:

(23) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(24) Obligation:

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(26) Obligation:

(27) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(28) Obligation:

(29) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(30) Obligation:

(31) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(32) Obligation:

(33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(34) Obligation:

(35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

(36) Obligation:

(37) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(38) Obligation:

(39) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

(40) BOUNDS(O(1), O(n^1))