Runtime Complexity (innermost) proof of /tmp/tmpofWPzK/LengthOfFiniteLists_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), 0 ms)

    ↳4 CdtProblem

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

    ↳6 CdtProblem

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

    ↳8 CdtProblem

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

    ↳10 CdtProblem

    ↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 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), 0 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), 11 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 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳42 CdtProblem

    ↳43 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 6 ms)

    ↳44 CdtProblem

↳45 CpxTrsMatchBoundsTAProof (⇔, 1023 ms)

↳46 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(cons(X1, X2)) → cons(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
length(mark(X)) → mark(length(X))
s(mark(X)) → mark(s(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
length(ok(X)) → ok(length(X))
s(ok(X)) → ok(s(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(zeros) → c(CONS(0, zeros))
ACTIVE(length(cons(z0, z1))) → c3(S(length(z1)), LENGTH(z1))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(zeros) → c(CONS(0, zeros))
ACTIVE(length(cons(z0, z1))) → c3(S(length(z1)), LENGTH(z1))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

ACTIVE, CONS, AND, LENGTH, S, PROPER, TOP

Compound Symbols:

c, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25

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

Removed 1 trailing nodes:

ACTIVE(zeros) → c(CONS(0, zeros))

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(length(cons(z0, z1))) → c3(S(length(z1)), LENGTH(z1))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(length(cons(z0, z1))) → c3(S(length(z1)), LENGTH(z1))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

ACTIVE, CONS, AND, LENGTH, S, PROPER, TOP

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25

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

Use narrowing to replace ACTIVE(length(cons(z0, z1))) → c3(S(length(z1)), LENGTH(z1)) by

ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(length(cons(x0, x1))) → c3

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(length(cons(x0, x1))) → c3

S tuples:

ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(length(cons(x0, x1))) → c3

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

ACTIVE, CONS, AND, LENGTH, S, PROPER, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25, c3, c3

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

Removed 1 trailing nodes:

ACTIVE(length(cons(x0, x1))) → c3

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))

S tuples:

ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

ACTIVE, CONS, AND, LENGTH, S, PROPER, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25, c3

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

Use narrowing to replace ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) by

ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1), ACTIVE(zeros))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(x0, x1)) → c4

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1), ACTIVE(zeros))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(x0, x1)) → c4

S tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1), ACTIVE(zeros))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(x0, x1)) → c4

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

ACTIVE, CONS, AND, LENGTH, S, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25, c3, c4, c4

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

Removed 1 trailing nodes:

ACTIVE(cons(x0, x1)) → c4

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1), ACTIVE(zeros))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))

S tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1), ACTIVE(zeros))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

ACTIVE, CONS, AND, LENGTH, S, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25, c3, c4

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

Removed 1 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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))

S tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

ACTIVE, CONS, AND, LENGTH, S, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25, c3, c4, c4

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

Use narrowing to replace ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) by

ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1), ACTIVE(zeros))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x0, x1)) → c5

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1), ACTIVE(zeros))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x0, x1)) → c5

S tuples:

ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1), ACTIVE(zeros))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x0, x1)) → c5

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

ACTIVE, CONS, AND, LENGTH, S, PROPER, TOP

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25, c3, c4, c4, c5, c5

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

Removed 1 trailing nodes:

ACTIVE(and(x0, x1)) → c5

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1), ACTIVE(zeros))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))

S tuples:

ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1), ACTIVE(zeros))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

ACTIVE, CONS, AND, LENGTH, S, PROPER, TOP

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25, c3, c4, c4, c5

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))

S tuples:

ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

ACTIVE, CONS, AND, LENGTH, S, PROPER, TOP

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25, c3, c4, c4, c5, c5

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

Use narrowing to replace ACTIVE(length(z0)) → c6(LENGTH(active(z0)), ACTIVE(z0)) by

ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))), ACTIVE(zeros))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)), ACTIVE(length(nil)))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(length(x0)) → c6

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))), ACTIVE(zeros))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)), ACTIVE(length(nil)))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(length(x0)) → c6

S tuples:

ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))), ACTIVE(zeros))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)), ACTIVE(length(nil)))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(length(x0)) → c6

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

ACTIVE, CONS, AND, LENGTH, S, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25, c3, c4, c4, c5, c5, c6, c6

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

Removed 1 trailing nodes:

ACTIVE(length(x0)) → c6

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))), ACTIVE(zeros))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)), ACTIVE(length(nil)))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))

S tuples:

ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))), ACTIVE(zeros))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)), ACTIVE(length(nil)))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

ACTIVE, CONS, AND, LENGTH, S, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25, c3, c4, c4, c5, c5, c6

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

Removed 5 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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))

S tuples:

ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

ACTIVE, CONS, AND, LENGTH, S, PROPER, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25, c3, c4, c4, c5, c5, c6, c6

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

Use narrowing to replace ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))), ACTIVE(zeros))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)), ACTIVE(length(nil)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x0)) → c7

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))), ACTIVE(zeros))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)), ACTIVE(length(nil)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x0)) → c7

S tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))), ACTIVE(zeros))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)), ACTIVE(length(nil)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x0)) → c7

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

CONS, AND, LENGTH, S, PROPER, TOP, ACTIVE

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25, c3, c4, c4, c5, c5, c6, c6, c7, c7

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

Removed 1 trailing nodes:

ACTIVE(s(x0)) → c7

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))), ACTIVE(zeros))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)), ACTIVE(length(nil)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))

S tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))), ACTIVE(zeros))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)), ACTIVE(length(nil)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

CONS, AND, LENGTH, S, PROPER, TOP, ACTIVE

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25, c3, c4, c4, c5, c5, c6, c6, c7

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

Removed 3 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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)))

S tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

CONS, AND, LENGTH, S, PROPER, TOP, ACTIVE

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c17, c19, c21, c23, c24, c25, c3, c4, c4, c5, c5, c6, c6, c7, c7

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

Use narrowing to replace PROPER(cons(z0, z1)) → c17(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, zeros)) → c17(CONS(proper(x0), ok(zeros)), PROPER(x0), PROPER(zeros))
PROPER(cons(x0, cons(z0, z1))) → c17(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, 0)) → c17(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, and(z0, z1))) → c17(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(cons(x0, tt)) → c17(CONS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(cons(x0, length(z0))) → c17(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, nil)) → c17(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, s(z0))) → c17(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(zeros, x1)) → c17(CONS(ok(zeros), proper(x1)), PROPER(zeros), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c17(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c17(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(and(z0, z1), x1)) → c17(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(cons(tt, x1)) → c17(CONS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(cons(length(z0), x1)) → c17(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(nil, x1)) → c17(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(s(z0), x1)) → c17(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(x0, x1)) → c17

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)))
PROPER(cons(x0, zeros)) → c17(CONS(proper(x0), ok(zeros)), PROPER(x0), PROPER(zeros))
PROPER(cons(x0, cons(z0, z1))) → c17(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, 0)) → c17(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, and(z0, z1))) → c17(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(cons(x0, tt)) → c17(CONS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(cons(x0, length(z0))) → c17(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, nil)) → c17(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, s(z0))) → c17(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(zeros, x1)) → c17(CONS(ok(zeros), proper(x1)), PROPER(zeros), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c17(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c17(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(and(z0, z1), x1)) → c17(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(cons(tt, x1)) → c17(CONS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(cons(length(z0), x1)) → c17(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(nil, x1)) → c17(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(s(z0), x1)) → c17(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(x0, x1)) → c17

S tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)))
PROPER(cons(x0, zeros)) → c17(CONS(proper(x0), ok(zeros)), PROPER(x0), PROPER(zeros))
PROPER(cons(x0, cons(z0, z1))) → c17(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, 0)) → c17(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, and(z0, z1))) → c17(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(cons(x0, tt)) → c17(CONS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(cons(x0, length(z0))) → c17(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, nil)) → c17(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, s(z0))) → c17(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(zeros, x1)) → c17(CONS(ok(zeros), proper(x1)), PROPER(zeros), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c17(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c17(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(and(z0, z1), x1)) → c17(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(cons(tt, x1)) → c17(CONS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(cons(length(z0), x1)) → c17(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(nil, x1)) → c17(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(s(z0), x1)) → c17(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(x0, x1)) → c17

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

CONS, AND, LENGTH, S, PROPER, TOP, ACTIVE

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c19, c21, c23, c24, c25, c3, c4, c4, c5, c5, c6, c6, c7, c7, c17, c17

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

Removed 1 trailing nodes:

PROPER(cons(x0, x1)) → c17

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)))
PROPER(cons(x0, zeros)) → c17(CONS(proper(x0), ok(zeros)), PROPER(x0), PROPER(zeros))
PROPER(cons(x0, cons(z0, z1))) → c17(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, 0)) → c17(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, and(z0, z1))) → c17(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(cons(x0, tt)) → c17(CONS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(cons(x0, length(z0))) → c17(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, nil)) → c17(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, s(z0))) → c17(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(zeros, x1)) → c17(CONS(ok(zeros), proper(x1)), PROPER(zeros), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c17(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c17(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(and(z0, z1), x1)) → c17(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(cons(tt, x1)) → c17(CONS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(cons(length(z0), x1)) → c17(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(nil, x1)) → c17(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(s(z0), x1)) → c17(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

S tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)))
PROPER(cons(x0, zeros)) → c17(CONS(proper(x0), ok(zeros)), PROPER(x0), PROPER(zeros))
PROPER(cons(x0, cons(z0, z1))) → c17(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, 0)) → c17(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, and(z0, z1))) → c17(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(cons(x0, tt)) → c17(CONS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(cons(x0, length(z0))) → c17(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, nil)) → c17(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, s(z0))) → c17(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(zeros, x1)) → c17(CONS(ok(zeros), proper(x1)), PROPER(zeros), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c17(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c17(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(and(z0, z1), x1)) → c17(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(cons(tt, x1)) → c17(CONS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(cons(length(z0), x1)) → c17(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(nil, x1)) → c17(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(s(z0), x1)) → c17(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

CONS, AND, LENGTH, S, PROPER, TOP, ACTIVE

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c19, c21, c23, c24, c25, c3, c4, c4, c5, c5, c6, c6, c7, c7, c17

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

Removed 8 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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)))
PROPER(cons(x0, cons(z0, z1))) → c17(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, and(z0, z1))) → c17(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(cons(x0, length(z0))) → c17(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, s(z0))) → c17(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(cons(z0, z1), x1)) → c17(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(and(z0, z1), x1)) → c17(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c17(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c17(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(x0, zeros)) → c17(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c17(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, tt)) → c17(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c17(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(zeros, x1)) → c17(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c17(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(tt, x1)) → c17(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c17(CONS(ok(nil), proper(x1)), PROPER(x1))

S tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)))
PROPER(cons(x0, cons(z0, z1))) → c17(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, and(z0, z1))) → c17(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(cons(x0, length(z0))) → c17(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, s(z0))) → c17(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(cons(z0, z1), x1)) → c17(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(and(z0, z1), x1)) → c17(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c17(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c17(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(x0, zeros)) → c17(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c17(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, tt)) → c17(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c17(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(zeros, x1)) → c17(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c17(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(tt, x1)) → c17(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c17(CONS(ok(nil), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

CONS, AND, LENGTH, S, PROPER, TOP, ACTIVE

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c19, c21, c23, c24, c25, c3, c4, c4, c5, c5, c6, c6, c7, c7, c17, c17

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

Use narrowing to replace PROPER(and(z0, z1)) → c19(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(and(x0, zeros)) → c19(AND(proper(x0), ok(zeros)), PROPER(x0), PROPER(zeros))
PROPER(and(x0, cons(z0, z1))) → c19(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(and(x0, 0)) → c19(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, and(z0, z1))) → c19(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c19(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, length(z0))) → c19(AND(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(and(x0, nil)) → c19(AND(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(and(x0, s(z0))) → c19(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(zeros, x1)) → c19(AND(ok(zeros), proper(x1)), PROPER(zeros), PROPER(x1))
PROPER(and(cons(z0, z1), x1)) → c19(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c19(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c19(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c19(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(length(z0), x1)) → c19(AND(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(and(nil, x1)) → c19(AND(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(and(s(z0), x1)) → c19(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, x1)) → c19

(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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)))
PROPER(cons(x0, cons(z0, z1))) → c17(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, and(z0, z1))) → c17(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(cons(x0, length(z0))) → c17(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, s(z0))) → c17(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(cons(z0, z1), x1)) → c17(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(and(z0, z1), x1)) → c17(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c17(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c17(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(x0, zeros)) → c17(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c17(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, tt)) → c17(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c17(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(zeros, x1)) → c17(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c17(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(tt, x1)) → c17(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c17(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(x0, zeros)) → c19(AND(proper(x0), ok(zeros)), PROPER(x0), PROPER(zeros))
PROPER(and(x0, cons(z0, z1))) → c19(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(and(x0, 0)) → c19(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, and(z0, z1))) → c19(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c19(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, length(z0))) → c19(AND(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(and(x0, nil)) → c19(AND(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(and(x0, s(z0))) → c19(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(zeros, x1)) → c19(AND(ok(zeros), proper(x1)), PROPER(zeros), PROPER(x1))
PROPER(and(cons(z0, z1), x1)) → c19(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c19(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c19(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c19(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(length(z0), x1)) → c19(AND(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(and(nil, x1)) → c19(AND(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(and(s(z0), x1)) → c19(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, x1)) → c19

S tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)))
PROPER(cons(x0, cons(z0, z1))) → c17(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, and(z0, z1))) → c17(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(cons(x0, length(z0))) → c17(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, s(z0))) → c17(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(cons(z0, z1), x1)) → c17(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(and(z0, z1), x1)) → c17(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c17(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c17(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(x0, zeros)) → c17(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c17(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, tt)) → c17(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c17(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(zeros, x1)) → c17(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c17(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(tt, x1)) → c17(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c17(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(x0, zeros)) → c19(AND(proper(x0), ok(zeros)), PROPER(x0), PROPER(zeros))
PROPER(and(x0, cons(z0, z1))) → c19(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(and(x0, 0)) → c19(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, and(z0, z1))) → c19(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c19(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, length(z0))) → c19(AND(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(and(x0, nil)) → c19(AND(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(and(x0, s(z0))) → c19(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(zeros, x1)) → c19(AND(ok(zeros), proper(x1)), PROPER(zeros), PROPER(x1))
PROPER(and(cons(z0, z1), x1)) → c19(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c19(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c19(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c19(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(length(z0), x1)) → c19(AND(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(and(nil, x1)) → c19(AND(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(and(s(z0), x1)) → c19(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, x1)) → c19

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

CONS, AND, LENGTH, S, PROPER, TOP, ACTIVE

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c21, c23, c24, c25, c3, c4, c4, c5, c5, c6, c6, c7, c7, c17, c17, c19, c19

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

Removed 1 trailing nodes:

PROPER(and(x0, x1)) → c19

(42) 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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)))
PROPER(cons(x0, cons(z0, z1))) → c17(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, and(z0, z1))) → c17(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(cons(x0, length(z0))) → c17(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, s(z0))) → c17(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(cons(z0, z1), x1)) → c17(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(and(z0, z1), x1)) → c17(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c17(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c17(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(x0, zeros)) → c17(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c17(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, tt)) → c17(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c17(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(zeros, x1)) → c17(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c17(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(tt, x1)) → c17(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c17(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(x0, zeros)) → c19(AND(proper(x0), ok(zeros)), PROPER(x0), PROPER(zeros))
PROPER(and(x0, cons(z0, z1))) → c19(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(and(x0, 0)) → c19(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, and(z0, z1))) → c19(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c19(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, length(z0))) → c19(AND(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(and(x0, nil)) → c19(AND(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(and(x0, s(z0))) → c19(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(zeros, x1)) → c19(AND(ok(zeros), proper(x1)), PROPER(zeros), PROPER(x1))
PROPER(and(cons(z0, z1), x1)) → c19(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c19(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c19(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c19(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(length(z0), x1)) → c19(AND(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(and(nil, x1)) → c19(AND(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(and(s(z0), x1)) → c19(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

S tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)))
PROPER(cons(x0, cons(z0, z1))) → c17(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, and(z0, z1))) → c17(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(cons(x0, length(z0))) → c17(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, s(z0))) → c17(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(cons(z0, z1), x1)) → c17(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(and(z0, z1), x1)) → c17(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c17(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c17(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(x0, zeros)) → c17(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c17(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, tt)) → c17(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c17(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(zeros, x1)) → c17(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c17(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(tt, x1)) → c17(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c17(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(x0, zeros)) → c19(AND(proper(x0), ok(zeros)), PROPER(x0), PROPER(zeros))
PROPER(and(x0, cons(z0, z1))) → c19(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(and(x0, 0)) → c19(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, and(z0, z1))) → c19(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c19(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, length(z0))) → c19(AND(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(and(x0, nil)) → c19(AND(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(and(x0, s(z0))) → c19(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(zeros, x1)) → c19(AND(ok(zeros), proper(x1)), PROPER(zeros), PROPER(x1))
PROPER(and(cons(z0, z1), x1)) → c19(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c19(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c19(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c19(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(length(z0), x1)) → c19(AND(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(and(nil, x1)) → c19(AND(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(and(s(z0), x1)) → c19(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

CONS, AND, LENGTH, S, PROPER, TOP, ACTIVE

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c21, c23, c24, c25, c3, c4, c4, c5, c5, c6, c6, c7, c7, c17, c17, c19

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

Removed 8 trailing tuple parts

(44) 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(cons(z0, z1)) → cons(active(z0), z1)
active(and(z0, z1)) → and(active(z0), z1)
active(length(z0)) → length(active(z0))
active(s(z0)) → s(active(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(zeros) → ok(zeros)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(0) → ok(0)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)))
PROPER(cons(x0, cons(z0, z1))) → c17(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, and(z0, z1))) → c17(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(cons(x0, length(z0))) → c17(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, s(z0))) → c17(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(cons(z0, z1), x1)) → c17(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(and(z0, z1), x1)) → c17(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c17(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c17(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(x0, zeros)) → c17(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c17(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, tt)) → c17(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c17(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(zeros, x1)) → c17(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c17(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(tt, x1)) → c17(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c17(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(x0, cons(z0, z1))) → c19(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c19(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, length(z0))) → c19(AND(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(and(x0, s(z0))) → c19(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(cons(z0, z1), x1)) → c19(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c19(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(length(z0), x1)) → c19(AND(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(and(s(z0), x1)) → c19(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, zeros)) → c19(AND(proper(x0), ok(zeros)), PROPER(x0))
PROPER(and(x0, 0)) → c19(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(x0, tt)) → c19(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, nil)) → c19(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(zeros, x1)) → c19(AND(ok(zeros), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c19(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c19(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(nil, x1)) → c19(AND(ok(nil), proper(x1)), PROPER(x1))

S tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
AND(mark(z0), z1) → c10(AND(z0, z1))
AND(ok(z0), ok(z1)) → c11(AND(z0, z1))
LENGTH(mark(z0)) → c12(LENGTH(z0))
LENGTH(ok(z0)) → c13(LENGTH(z0))
S(mark(z0)) → c14(S(z0))
S(ok(z0)) → c15(S(z0))
PROPER(length(z0)) → c21(LENGTH(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c24(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c25(TOP(active(z0)), ACTIVE(z0))
ACTIVE(length(cons(x0, mark(z0)))) → c3(S(mark(length(z0))), LENGTH(mark(z0)))
ACTIVE(length(cons(x0, ok(z0)))) → c3(S(ok(length(z0))), LENGTH(ok(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c4(CONS(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(and(z0, z1), x1)) → c4(CONS(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(cons(length(z0), x1)) → c4(CONS(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(zeros, x1)) → c4(CONS(mark(cons(0, zeros)), x1))
ACTIVE(and(length(cons(z0, z1)), x1)) → c5(AND(mark(s(length(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(and(cons(z0, z1), x1)) → c5(AND(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(length(z0), x1)) → c5(AND(length(active(z0)), x1), ACTIVE(length(z0)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(and(tt, z0), x1)) → c4(CONS(mark(z0), x1))
ACTIVE(and(zeros, x1)) → c5(AND(mark(cons(0, zeros)), x1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(length(length(cons(z0, z1)))) → c6(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(cons(z0, z1))) → c6(LENGTH(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(length(and(z0, z1))) → c6(LENGTH(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(length(length(z0))) → c6(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(length(s(z0))) → c6(LENGTH(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c4(CONS(mark(0), x1))
ACTIVE(and(length(nil), x1)) → c5(AND(mark(0), x1))
ACTIVE(length(zeros)) → c6(LENGTH(mark(cons(0, zeros))))
ACTIVE(length(and(tt, z0))) → c6(LENGTH(mark(z0)))
ACTIVE(length(length(nil))) → c6(LENGTH(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c7(S(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(and(z0, z1))) → c7(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(length(z0))) → c7(S(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(zeros)) → c7(S(mark(cons(0, zeros))))
ACTIVE(s(and(tt, z0))) → c7(S(mark(z0)))
ACTIVE(s(length(nil))) → c7(S(mark(0)))
PROPER(cons(x0, cons(z0, z1))) → c17(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, and(z0, z1))) → c17(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(cons(x0, length(z0))) → c17(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, s(z0))) → c17(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(cons(z0, z1), x1)) → c17(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(and(z0, z1), x1)) → c17(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c17(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c17(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(x0, zeros)) → c17(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c17(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, tt)) → c17(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c17(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(zeros, x1)) → c17(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c17(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(tt, x1)) → c17(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c17(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(x0, cons(z0, z1))) → c19(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c19(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, length(z0))) → c19(AND(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(and(x0, s(z0))) → c19(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(cons(z0, z1), x1)) → c19(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c19(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(length(z0), x1)) → c19(AND(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(and(s(z0), x1)) → c19(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, zeros)) → c19(AND(proper(x0), ok(zeros)), PROPER(x0))
PROPER(and(x0, 0)) → c19(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(x0, tt)) → c19(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, nil)) → c19(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(zeros, x1)) → c19(AND(ok(zeros), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c19(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c19(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(nil, x1)) → c19(AND(ok(nil), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, cons, and, length, s, proper, top

Defined Pair Symbols:

CONS, AND, LENGTH, S, PROPER, TOP, ACTIVE

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c21, c23, c24, c25, c3, c4, c4, c5, c5, c6, c6, c7, c7, c17, c17, c19, c19

(45) 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 5.

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
mark0(0) → 0
00() → 0
tt0() → 0
nil0() → 0
ok0(0) → 0
active0(0) → 1
cons0(0, 0) → 2
and0(0, 0) → 3
length0(0) → 4
s0(0) → 5
proper0(0) → 6
top0(0) → 7
01() → 9
zeros1() → 10
cons1(9, 10) → 8
mark1(8) → 1
cons1(0, 0) → 11
mark1(11) → 2
and1(0, 0) → 12
mark1(12) → 3
length1(0) → 13
mark1(13) → 4
s1(0) → 14
mark1(14) → 5
zeros1() → 15
ok1(15) → 6
01() → 16
ok1(16) → 6
tt1() → 17
ok1(17) → 6
nil1() → 18
ok1(18) → 6
cons1(0, 0) → 19
ok1(19) → 2
and1(0, 0) → 20
ok1(20) → 3
length1(0) → 21
ok1(21) → 4
s1(0) → 22
ok1(22) → 5
proper1(0) → 23
top1(23) → 7
active1(0) → 24
top1(24) → 7
mark1(8) → 24
mark1(11) → 11
mark1(11) → 19
mark1(12) → 12
mark1(12) → 20
mark1(13) → 13
mark1(13) → 21
mark1(14) → 14
mark1(14) → 22
ok1(15) → 23
ok1(16) → 23
ok1(17) → 23
ok1(18) → 23
ok1(19) → 11
ok1(19) → 19
ok1(20) → 12
ok1(20) → 20
ok1(21) → 13
ok1(21) → 21
ok1(22) → 14
ok1(22) → 22
proper2(8) → 25
top2(25) → 7
active2(15) → 26
top2(26) → 7
active2(16) → 26
active2(17) → 26
active2(18) → 26
02() → 28
zeros2() → 29
cons2(28, 29) → 27
mark2(27) → 26
proper2(9) → 30
proper2(10) → 31
cons2(30, 31) → 25
zeros2() → 32
ok2(32) → 31
02() → 33
ok2(33) → 30
proper3(27) → 34
top3(34) → 7
proper3(28) → 35
proper3(29) → 36
cons3(35, 36) → 34
cons3(33, 32) → 37
ok3(37) → 25
zeros3() → 38
ok3(38) → 36
03() → 39
ok3(39) → 35
active3(37) → 40
top3(40) → 7
cons4(39, 38) → 41
ok4(41) → 34
active4(33) → 42
cons4(42, 32) → 40
active4(41) → 43
top4(43) → 7
active5(39) → 44
cons5(44, 38) → 43

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

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

(42) Obligation:

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

(44) Obligation:

(45) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

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