Runtime Complexity (innermost) proof of /tmp/tmpAMHgWz/OvConsOS_nosorts

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), 0 ms)

↳2 CdtProblem

    ↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 1 ms)

    ↳4 CdtProblem

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

    ↳6 CdtProblem

    ↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 51 ms)

    ↳8 CdtProblem

    ↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 7 ms)

    ↳10 CdtProblem

    ↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 32 ms)

    ↳12 CdtProblem

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

    ↳14 CdtProblem

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

    ↳16 CdtProblem

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

    ↳18 CdtProblem

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

    ↳20 CdtProblem

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

    ↳22 CdtProblem

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

    ↳24 CdtProblem

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

    ↳26 CdtProblem

    ↳27 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 53 ms)

    ↳28 CdtProblem

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

    ↳30 CdtProblem

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

    ↳32 CdtProblem

    ↳33 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 18 ms)

    ↳34 CdtProblem

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

    ↳36 CdtProblem

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

    ↳38 CdtProblem

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

    ↳40 CdtProblem

↳41 CpxTrsMatchBoundsTAProof (⇔, 217 ms)

↳42 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(and(tt, X)) → mark(X)
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(s(length(L)))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(cons(N, take(M, IL)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
mark(s(X)) → active(s(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
length(mark(X)) → length(X)
length(active(X)) → length(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(nil)) → c2(MARK(0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
ACTIVE(take(0, z0)) → c4(MARK(nil))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(cons(z0, z1)) → c7(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(0) → c8(ACTIVE(0))
MARK(and(z0, z1)) → c9(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(tt) → c10(ACTIVE(tt))
MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(nil) → c12(ACTIVE(nil))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))

S tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(nil)) → c2(MARK(0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
ACTIVE(take(0, z0)) → c4(MARK(nil))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(cons(z0, z1)) → c7(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(0) → c8(ACTIVE(0))
MARK(and(z0, z1)) → c9(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(tt) → c10(ACTIVE(tt))
MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(nil) → c12(ACTIVE(nil))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30

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

Removed 5 trailing nodes:

ACTIVE(take(0, z0)) → c4(MARK(nil))
MARK(nil) → c12(ACTIVE(nil))
ACTIVE(length(nil)) → c2(MARK(0))
MARK(0) → c8(ACTIVE(0))
MARK(tt) → c10(ACTIVE(tt))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(cons(z0, z1)) → c7(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c9(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))

S tuples:

ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(cons(z0, z1)) → c7(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c9(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c, c1, c3, c5, c6, c7, c9, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30

(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(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(cons(z0, z1)) → c7(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c9(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(cons(z0, z1)) → c7(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c9(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c7, c9, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3

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

Use narrowing to replace MARK(cons(z0, z1)) → c7(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0)) by

MARK(cons(z0, z1)) → c7(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(and(z0, z1)) → c9(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(z0, z1)) → c7(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(and(z0, z1)) → c9(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(z0, z1)) → c7(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c9, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7

(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(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(and(z0, z1)) → c9(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(and(z0, z1)) → c9(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c9, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7

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

Use narrowing to replace MARK(and(z0, z1)) → c9(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0)) by

MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1), MARK(0))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1), MARK(tt))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1), MARK(nil))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1), MARK(0))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1), MARK(tt))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1), MARK(nil))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1), MARK(0))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1), MARK(tt))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1), MARK(nil))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9

(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(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9, c9

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

Use narrowing to replace MARK(length(z0)) → c11(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0)) by

MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(x0)) → c11

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(x0)) → c11

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(x0)) → c11

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9, c9, c11, c11

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

Removed 1 trailing nodes:

MARK(length(x0)) → c11

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9, c9, c11

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

Removed 3 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9, c9, c11, c11

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

Use narrowing to replace MARK(s(z0)) → c13(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0)) by

MARK(s(z0)) → c13(ACTIVE(s(z0)), S(mark(z0)), MARK(z0))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)), MARK(0))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)), MARK(tt))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)), MARK(nil))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(x0)) → c13

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(z0)) → c13(ACTIVE(s(z0)), S(mark(z0)), MARK(z0))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)), MARK(0))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)), MARK(tt))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)), MARK(nil))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(x0)) → c13

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(z0)) → c13(ACTIVE(s(z0)), S(mark(z0)), MARK(z0))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)), MARK(0))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)), MARK(tt))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)), MARK(nil))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(x0)) → c13

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9, c9, c11, c11, c13, c13

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

Removed 1 trailing nodes:

MARK(s(x0)) → c13

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(z0)) → c13(ACTIVE(s(z0)), S(mark(z0)), MARK(z0))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)), MARK(0))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)), MARK(tt))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)), MARK(nil))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(z0)) → c13(ACTIVE(s(z0)), S(mark(z0)), MARK(z0))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)), MARK(0))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)), MARK(tt))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)), MARK(nil))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9, c9, c11, c11, c13

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

Removed 4 trailing tuple parts

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9, c9, c11, c11, c13, c13

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

Use narrowing to replace MARK(take(z0, z1)) → c14(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1)) by

MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0), MARK(0))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0), MARK(tt))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0), MARK(nil))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(0), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(tt), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(nil), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))
MARK(take(x0, x1)) → c14

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0), MARK(0))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0), MARK(tt))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0), MARK(nil))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(0), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(tt), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(nil), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))
MARK(take(x0, x1)) → c14

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0), MARK(0))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0), MARK(tt))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0), MARK(nil))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(0), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(tt), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(nil), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))
MARK(take(x0, x1)) → c14

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9, c9, c11, c11, c13, c13, c14, c14

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

Removed 1 trailing nodes:

MARK(take(x0, x1)) → c14

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0), MARK(0))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0), MARK(tt))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0), MARK(nil))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(0), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(tt), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(nil), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0), MARK(0))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0), MARK(tt))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0), MARK(nil))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(0), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(tt), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(nil), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9, c9, c11, c11, c13, c13, c14

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

Removed 6 trailing tuple parts

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(x1))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(x1))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9, c9, c11, c11, c13, c13, c14, c14

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

Use narrowing to replace MARK(cons(zeros, x1)) → c7(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros)) by

MARK(cons(zeros, z1)) → c7(ACTIVE(cons(zeros, z1)), CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(zeros, x0)) → c7(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c7

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(x1))
MARK(cons(zeros, z1)) → c7(ACTIVE(cons(zeros, z1)), CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(zeros, x0)) → c7(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c7

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(x1))
MARK(cons(zeros, z1)) → c7(ACTIVE(cons(zeros, z1)), CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(zeros, x0)) → c7(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, x0)) → c7

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9, c9, c11, c11, c13, c13, c14, c14, c7

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

Removed 1 trailing nodes:

MARK(cons(zeros, x0)) → c7

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(x1))
MARK(cons(zeros, z1)) → c7(ACTIVE(cons(zeros, z1)), CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(zeros, x0)) → c7(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(x1))
MARK(cons(zeros, z1)) → c7(ACTIVE(cons(zeros, z1)), CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(zeros, x0)) → c7(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9, c9, c11, c11, c13, c13, c14, c14

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

Removed 1 trailing tuple parts

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(x1))
MARK(cons(zeros, x0)) → c7(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, z1)) → c7(CONS(mark(zeros), z1), MARK(zeros))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(cons(z0, z1), x1)) → c7(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(x1))
MARK(cons(zeros, x0)) → c7(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, z1)) → c7(CONS(mark(zeros), z1), MARK(zeros))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9, c9, c11, c11, c13, c13, c14, c14

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

Use narrowing to replace MARK(cons(cons(z0, z1), x1)) → c7(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)) → c7(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c7(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c7(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)) → c7(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)) → c7(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(and(z0, z1), x1), x2)) → c7(ACTIVE(cons(active(cons(active(and(mark(z0), z1)), x1)), x2)), CONS(mark(cons(and(z0, z1), x1)), x2), MARK(cons(and(z0, z1), x1)))
MARK(cons(cons(tt, x1), x2)) → c7(ACTIVE(cons(active(cons(active(tt), x1)), x2)), CONS(mark(cons(tt, x1)), x2), MARK(cons(tt, x1)))
MARK(cons(cons(length(z0), x1), x2)) → c7(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)) → c7(ACTIVE(cons(active(cons(active(nil), x1)), x2)), CONS(mark(cons(nil, x1)), x2), MARK(cons(nil, x1)))
MARK(cons(cons(s(z0), x1), x2)) → c7(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(take(z0, z1), x1), x2)) → c7(ACTIVE(cons(active(cons(active(take(mark(z0), mark(z1))), x1)), x2)), CONS(mark(cons(take(z0, z1), x1)), x2), MARK(cons(take(z0, z1), x1)))
MARK(cons(cons(x0, x1), x2)) → c7(CONS(mark(cons(x0, x1)), x2))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
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)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)

Tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(x1))
MARK(cons(zeros, x0)) → c7(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, z1)) → c7(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c7(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c7(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c7(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)) → c7(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)) → c7(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(and(z0, z1), x1), x2)) → c7(ACTIVE(cons(active(cons(active(and(mark(z0), z1)), x1)), x2)), CONS(mark(cons(and(z0, z1), x1)), x2), MARK(cons(and(z0, z1), x1)))
MARK(cons(cons(tt, x1), x2)) → c7(ACTIVE(cons(active(cons(active(tt), x1)), x2)), CONS(mark(cons(tt, x1)), x2), MARK(cons(tt, x1)))
MARK(cons(cons(length(z0), x1), x2)) → c7(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)) → c7(ACTIVE(cons(active(cons(active(nil), x1)), x2)), CONS(mark(cons(nil, x1)), x2), MARK(cons(nil, x1)))
MARK(cons(cons(s(z0), x1), x2)) → c7(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(take(z0, z1), x1), x2)) → c7(ACTIVE(cons(active(cons(active(take(mark(z0), mark(z1))), x1)), x2)), CONS(mark(cons(take(z0, z1), x1)), x2), MARK(cons(take(z0, z1), x1)))
MARK(cons(cons(x0, x1), x2)) → c7(CONS(mark(cons(x0, x1)), x2))

S tuples:

ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
MARK(zeros) → c6(ACTIVE(zeros))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
CONS(z0, mark(z1)) → c16(CONS(z0, z1))
CONS(active(z0), z1) → c17(CONS(z0, z1))
CONS(z0, active(z1)) → c18(CONS(z0, z1))
AND(mark(z0), z1) → c19(AND(z0, z1))
AND(z0, mark(z1)) → c20(AND(z0, z1))
AND(active(z0), z1) → c21(AND(z0, z1))
AND(z0, active(z1)) → c22(AND(z0, z1))
LENGTH(mark(z0)) → c23(LENGTH(z0))
LENGTH(active(z0)) → c24(LENGTH(z0))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
TAKE(mark(z0), z1) → c27(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c28(TAKE(z0, z1))
TAKE(active(z0), z1) → c29(TAKE(z0, z1))
TAKE(z0, active(z1)) → c30(TAKE(z0, z1))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), LENGTH(z1))
MARK(cons(and(z0, z1), x1)) → c7(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(length(z0), x1)) → c7(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c7(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(take(z0, z1), x1)) → c7(ACTIVE(cons(active(take(mark(z0), mark(z1))), x1)), CONS(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(cons(x0, x1)) → c7(CONS(mark(x0), x1))
MARK(cons(z0, z1)) → c7(CONS(mark(z0), z1), MARK(z0))
MARK(cons(0, x1)) → c7(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1))
MARK(cons(tt, x1)) → c7(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1))
MARK(cons(nil, x1)) → c7(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1))
MARK(and(z0, z1)) → c9(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c9(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c9(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(and(z0, z1), x1)) → c9(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(length(z0), x1)) → c9(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(s(z0), x1)) → c9(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(take(z0, z1), x1)) → c9(ACTIVE(and(active(take(mark(z0), mark(z1))), x1)), AND(mark(take(z0, z1)), x1), MARK(take(z0, z1)))
MARK(and(x0, x1)) → c9(AND(mark(x0), x1))
MARK(and(0, x1)) → c9(ACTIVE(and(active(0), x1)), AND(mark(0), x1))
MARK(and(tt, x1)) → c9(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1))
MARK(and(nil, x1)) → c9(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1))
MARK(length(z0)) → c11(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c11(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c11(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(and(z0, z1))) → c11(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c11(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c11(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(take(z0, z1))) → c11(ACTIVE(length(active(take(mark(z0), mark(z1))))), LENGTH(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(length(0)) → c11(ACTIVE(length(active(0))), LENGTH(mark(0)))
MARK(length(tt)) → c11(ACTIVE(length(active(tt))), LENGTH(mark(tt)))
MARK(length(nil)) → c11(ACTIVE(length(active(nil))), LENGTH(mark(nil)))
MARK(s(zeros)) → c13(ACTIVE(s(active(zeros))), S(mark(zeros)), MARK(zeros))
MARK(s(cons(z0, z1))) → c13(ACTIVE(s(active(cons(mark(z0), z1)))), S(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(s(and(z0, z1))) → c13(ACTIVE(s(active(and(mark(z0), z1)))), S(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(s(length(z0))) → c13(ACTIVE(s(active(length(mark(z0))))), S(mark(length(z0))), MARK(length(z0)))
MARK(s(s(z0))) → c13(ACTIVE(s(active(s(mark(z0))))), S(mark(s(z0))), MARK(s(z0)))
MARK(s(take(z0, z1))) → c13(ACTIVE(s(active(take(mark(z0), mark(z1))))), S(mark(take(z0, z1))), MARK(take(z0, z1)))
MARK(s(z0)) → c13(S(mark(z0)), MARK(z0))
MARK(s(0)) → c13(ACTIVE(s(active(0))), S(mark(0)))
MARK(s(tt)) → c13(ACTIVE(s(active(tt))), S(mark(tt)))
MARK(s(nil)) → c13(ACTIVE(s(active(nil))), S(mark(nil)))
MARK(take(z0, x1)) → c14(ACTIVE(take(z0, mark(x1))), TAKE(mark(z0), mark(x1)), MARK(z0), MARK(x1))
MARK(take(x0, z1)) → c14(ACTIVE(take(mark(x0), z1)), TAKE(mark(x0), mark(z1)), MARK(x0), MARK(z1))
MARK(take(x0, zeros)) → c14(ACTIVE(take(mark(x0), active(zeros))), TAKE(mark(x0), mark(zeros)), MARK(x0), MARK(zeros))
MARK(take(x0, cons(z0, z1))) → c14(ACTIVE(take(mark(x0), active(cons(mark(z0), z1)))), TAKE(mark(x0), mark(cons(z0, z1))), MARK(x0), MARK(cons(z0, z1)))
MARK(take(x0, and(z0, z1))) → c14(ACTIVE(take(mark(x0), active(and(mark(z0), z1)))), TAKE(mark(x0), mark(and(z0, z1))), MARK(x0), MARK(and(z0, z1)))
MARK(take(x0, length(z0))) → c14(ACTIVE(take(mark(x0), active(length(mark(z0))))), TAKE(mark(x0), mark(length(z0))), MARK(x0), MARK(length(z0)))
MARK(take(x0, s(z0))) → c14(ACTIVE(take(mark(x0), active(s(mark(z0))))), TAKE(mark(x0), mark(s(z0))), MARK(x0), MARK(s(z0)))
MARK(take(x0, take(z0, z1))) → c14(ACTIVE(take(mark(x0), active(take(mark(z0), mark(z1))))), TAKE(mark(x0), mark(take(z0, z1))), MARK(x0), MARK(take(z0, z1)))
MARK(take(zeros, x1)) → c14(ACTIVE(take(active(zeros), mark(x1))), TAKE(mark(zeros), mark(x1)), MARK(zeros), MARK(x1))
MARK(take(cons(z0, z1), x1)) → c14(ACTIVE(take(active(cons(mark(z0), z1)), mark(x1))), TAKE(mark(cons(z0, z1)), mark(x1)), MARK(cons(z0, z1)), MARK(x1))
MARK(take(and(z0, z1), x1)) → c14(ACTIVE(take(active(and(mark(z0), z1)), mark(x1))), TAKE(mark(and(z0, z1)), mark(x1)), MARK(and(z0, z1)), MARK(x1))
MARK(take(length(z0), x1)) → c14(ACTIVE(take(active(length(mark(z0))), mark(x1))), TAKE(mark(length(z0)), mark(x1)), MARK(length(z0)), MARK(x1))
MARK(take(s(z0), x1)) → c14(ACTIVE(take(active(s(mark(z0))), mark(x1))), TAKE(mark(s(z0)), mark(x1)), MARK(s(z0)), MARK(x1))
MARK(take(take(z0, z1), x1)) → c14(ACTIVE(take(active(take(mark(z0), mark(z1))), mark(x1))), TAKE(mark(take(z0, z1)), mark(x1)), MARK(take(z0, z1)), MARK(x1))
MARK(take(x0, 0)) → c14(ACTIVE(take(mark(x0), active(0))), TAKE(mark(x0), mark(0)), MARK(x0))
MARK(take(x0, tt)) → c14(ACTIVE(take(mark(x0), active(tt))), TAKE(mark(x0), mark(tt)), MARK(x0))
MARK(take(x0, nil)) → c14(ACTIVE(take(mark(x0), active(nil))), TAKE(mark(x0), mark(nil)), MARK(x0))
MARK(take(0, x1)) → c14(ACTIVE(take(active(0), mark(x1))), TAKE(mark(0), mark(x1)), MARK(x1))
MARK(take(tt, x1)) → c14(ACTIVE(take(active(tt), mark(x1))), TAKE(mark(tt), mark(x1)), MARK(x1))
MARK(take(nil, x1)) → c14(ACTIVE(take(active(nil), mark(x1))), TAKE(mark(nil), mark(x1)), MARK(x1))
MARK(cons(zeros, x0)) → c7(ACTIVE(cons(mark(cons(0, zeros)), x0)), CONS(mark(zeros), x0), MARK(zeros))
MARK(cons(zeros, z1)) → c7(CONS(mark(zeros), z1), MARK(zeros))
MARK(cons(cons(x0, x1), z1)) → c7(ACTIVE(cons(cons(mark(x0), x1), z1)), CONS(mark(cons(x0, x1)), z1), MARK(cons(x0, x1)))
MARK(cons(cons(z0, z1), x2)) → c7(ACTIVE(cons(active(cons(z0, z1)), x2)), CONS(mark(cons(z0, z1)), x2), MARK(cons(z0, z1)))
MARK(cons(cons(zeros, x1), x2)) → c7(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)) → c7(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)) → c7(ACTIVE(cons(active(cons(active(0), x1)), x2)), CONS(mark(cons(0, x1)), x2), MARK(cons(0, x1)))
MARK(cons(cons(and(z0, z1), x1), x2)) → c7(ACTIVE(cons(active(cons(active(and(mark(z0), z1)), x1)), x2)), CONS(mark(cons(and(z0, z1), x1)), x2), MARK(cons(and(z0, z1), x1)))
MARK(cons(cons(tt, x1), x2)) → c7(ACTIVE(cons(active(cons(active(tt), x1)), x2)), CONS(mark(cons(tt, x1)), x2), MARK(cons(tt, x1)))
MARK(cons(cons(length(z0), x1), x2)) → c7(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)) → c7(ACTIVE(cons(active(cons(active(nil), x1)), x2)), CONS(mark(cons(nil, x1)), x2), MARK(cons(nil, x1)))
MARK(cons(cons(s(z0), x1), x2)) → c7(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(take(z0, z1), x1), x2)) → c7(ACTIVE(cons(active(cons(active(take(mark(z0), mark(z1))), x1)), x2)), CONS(mark(cons(take(z0, z1), x1)), x2), MARK(cons(take(z0, z1), x1)))
MARK(cons(cons(x0, x1), x2)) → c7(CONS(mark(cons(x0, x1)), x2))

K tuples:none
Defined Rule Symbols:

active, mark, cons, and, length, s, take

Defined Pair Symbols:

ACTIVE, MARK, CONS, AND, LENGTH, S, TAKE

Compound Symbols:

c1, c5, c6, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c, c3, c7, c7, c7, c9, c9, c9, c11, c11, c13, c13, c14, c14

(41) 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
and0(0, 0) → 4
length0(0) → 5
s0(0) → 6
take0(0, 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) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

(41) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

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