Runtime Complexity (innermost) proof of /tmp/tmphymbiJ/Ex1_GL02a

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

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

↳4 CdtProblem

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

↳6 CdtProblem

↳7 CdtUnreachableProof (⇔, 3 ms)

↳8 CdtProblem

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

↳10 CdtProblem

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

↳12 CdtProblem

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

↳14 CdtProblem

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

↳16 CdtProblem

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

↳18 CdtProblem

↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 522 ms)

↳20 CdtProblem

↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 816 ms)

↳22 CdtProblem

↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 10 ms)

↳24 CdtProblem

↳25 CdtUnreachableProof (⇔, 0 ms)

↳26 CdtProblem

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

↳28 CdtProblem

↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 23 ms)

↳30 CdtProblem

↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

↳32 CdtProblem

↳33 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 10 ms)

↳34 CdtProblem

↳35 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

↳36 CdtProblem

↳37 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 0 ms)

↳38 CdtProblem

↳39 SIsEmptyProof (⇔, 0 ms)

↳40 BOUNDS(O(1), O(1))

(0) Obligation:

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

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(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(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(inf(z0)) → c3(CONS(z0, inf(s(z0))), INF(s(z0)), S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(inf(z0)) → c3(CONS(z0, inf(s(z0))), INF(s(z0)), S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c3, c5, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33

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

Removed 2 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))

K tuples:none
Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c5, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3

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

Use narrowing to replace ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2)) by

ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(CONS(x1, mark(take(z0, z1))), TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(CONS(x1, mark(take(z0, z1))), TAKE(z0, mark(z1)))
ACTIVE(take(s(ok(z0)), cons(x1, ok(z1)))) → c5(CONS(x1, ok(take(z0, z1))), TAKE(ok(z0), ok(z1)))
ACTIVE(take(s(x0), cons(x1, x2))) → c5

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(CONS(x1, mark(take(z0, z1))), TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(CONS(x1, mark(take(z0, z1))), TAKE(z0, mark(z1)))
ACTIVE(take(s(ok(z0)), cons(x1, ok(z1)))) → c5(CONS(x1, ok(take(z0, z1))), TAKE(ok(z0), ok(z1)))
ACTIVE(take(s(x0), cons(x1, x2))) → c5

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(CONS(x1, mark(take(z0, z1))), TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(CONS(x1, mark(take(z0, z1))), TAKE(z0, mark(z1)))
ACTIVE(take(s(ok(z0)), cons(x1, ok(z1)))) → c5(CONS(x1, ok(take(z0, z1))), TAKE(ok(z0), ok(z1)))
ACTIVE(take(s(x0), cons(x1, x2))) → c5

K tuples:none
Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c5, c5

(7) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(take(s(ok(z0)), cons(x1, ok(z1)))) → c5(CONS(x1, ok(take(z0, z1))), TAKE(ok(z0), ok(z1)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(CONS(x1, mark(take(z0, z1))), TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(CONS(x1, mark(take(z0, z1))), TAKE(z0, mark(z1)))
ACTIVE(take(s(x0), cons(x1, x2))) → c5
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(CONS(x1, mark(take(z0, z1))), TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(CONS(x1, mark(take(z0, z1))), TAKE(z0, mark(z1)))
ACTIVE(take(s(x0), cons(x1, x2))) → c5

K tuples:none
Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS, TOP, ACTIVE, PROPER

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c32, c33, c1, c7, c8, c9, c10, c11, c3, c5, c5, c19, c22, c24, c25, c26, c28

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

Removed 1 trailing nodes:

ACTIVE(take(s(x0), cons(x1, x2))) → c5

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(CONS(x1, mark(take(z0, z1))), TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(CONS(x1, mark(take(z0, z1))), TAKE(z0, mark(z1)))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(CONS(x1, mark(take(z0, z1))), TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(CONS(x1, mark(take(z0, z1))), TAKE(z0, mark(z1)))

K tuples:none
Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS, TOP, ACTIVE, PROPER

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c32, c33, c1, c7, c8, c9, c10, c11, c3, c5, c19, c22, c24, c25, c26, c28

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

Removed 2 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))

K tuples:none
Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS, TOP, ACTIVE, PROPER

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c32, c33, c1, c7, c8, c9, c10, c11, c3, c19, c22, c24, c25, c26, c28, c5

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

Use narrowing to replace TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)), PROPER(0))
TOP(mark(true)) → c32(TOP(ok(true)), PROPER(true))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(false)) → c32(TOP(ok(false)), PROPER(false))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(nil)) → c32(TOP(ok(nil)), PROPER(nil))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(x0)) → c32

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)), PROPER(0))
TOP(mark(true)) → c32(TOP(ok(true)), PROPER(true))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(false)) → c32(TOP(ok(false)), PROPER(false))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(nil)) → c32(TOP(ok(nil)), PROPER(nil))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(x0)) → c32

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)), PROPER(0))
TOP(mark(true)) → c32(TOP(ok(true)), PROPER(true))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(false)) → c32(TOP(ok(false)), PROPER(false))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(nil)) → c32(TOP(ok(nil)), PROPER(nil))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(x0)) → c32

K tuples:none
Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS, TOP, ACTIVE, PROPER

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c33, c1, c7, c8, c9, c10, c11, c3, c19, c22, c24, c25, c26, c28, c5, c32, c32

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c32

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)), PROPER(0))
TOP(mark(true)) → c32(TOP(ok(true)), PROPER(true))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(false)) → c32(TOP(ok(false)), PROPER(false))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(nil)) → c32(TOP(ok(nil)), PROPER(nil))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)), PROPER(0))
TOP(mark(true)) → c32(TOP(ok(true)), PROPER(true))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(false)) → c32(TOP(ok(false)), PROPER(false))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(nil)) → c32(TOP(ok(nil)), PROPER(nil))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))

K tuples:none
Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS, TOP, ACTIVE, PROPER

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c33, c1, c7, c8, c9, c10, c11, c3, c19, c22, c24, c25, c26, c28, c5, c32

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

Removed 4 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

K tuples:none
Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS, TOP, ACTIVE, PROPER

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c33, c1, c7, c8, c9, c10, c11, c3, c19, c22, c24, c25, c26, c28, c5, c32, c32

(19) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

We considered the (Usable) Rules:

proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
s(ok(z0)) → ok(s(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))

And the Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(CONS(x₁, x₂)) = 0
POL(EQ(x₁, x₂)) = 0
POL(INF(x₁)) = 0
POL(LENGTH(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TAKE(x₁, x₂)) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c24(x₁, x₂)) = x₁ + x₂
POL(c25(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c26(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c28(x₁, x₂)) = x₁ + x₂
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c32(x₁, x₂)) = x₁ + x₂
POL(c33(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 0
POL(eq(x₁, x₂)) = [2]
POL(false) = [2]
POL(inf(x₁)) = [2]
POL(length(x₁)) = [2]
POL(mark(x₁)) = [2]
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = [2]
POL(true) = [2]

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))

K tuples:

TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS, TOP, ACTIVE, PROPER

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c33, c1, c7, c8, c9, c10, c11, c3, c19, c22, c24, c25, c26, c28, c5, c32, c32

(21) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))

We considered the (Usable) Rules:

proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
s(ok(z0)) → ok(s(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))

And the Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(CONS(x₁, x₂)) = 0
POL(EQ(x₁, x₂)) = 0
POL(INF(x₁)) = 0
POL(LENGTH(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TAKE(x₁, x₂)) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c24(x₁, x₂)) = x₁ + x₂
POL(c25(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c26(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c28(x₁, x₂)) = x₁ + x₂
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c32(x₁, x₂)) = x₁ + x₂
POL(c33(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 0
POL(eq(x₁, x₂)) = [2]
POL(false) = 0
POL(inf(x₁)) = [2]
POL(length(x₁)) = [2]
POL(mark(x₁)) = [2]
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = [2]
POL(true) = 0

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))

K tuples:

TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(nil)) → c32(TOP(ok(nil)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))

Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS, TOP, ACTIVE, PROPER

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c33, c1, c7, c8, c9, c10, c11, c3, c19, c22, c24, c25, c26, c28, c5, c32, c32

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

Use narrowing to replace TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(eq(0, 0))) → c33(TOP(mark(true)), ACTIVE(eq(0, 0)))
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1)))
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0)))
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil)))
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0)))
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1)))
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1)))
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))
TOP(ok(x0)) → c33

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))
TOP(ok(eq(0, 0))) → c33(TOP(mark(true)), ACTIVE(eq(0, 0)))
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1)))
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0)))
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil)))
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0)))
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1)))
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1)))
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))
TOP(ok(x0)) → c33

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(ok(eq(0, 0))) → c33(TOP(mark(true)), ACTIVE(eq(0, 0)))
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1)))
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0)))
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil)))
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0)))
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1)))
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1)))
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))
TOP(ok(x0)) → c33

K tuples:

TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(nil)) → c32(TOP(ok(nil)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))

Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS, ACTIVE, PROPER, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c1, c7, c8, c9, c10, c11, c3, c19, c22, c24, c25, c26, c28, c5, c32, c32, c33, c33

(25) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
ACTIVE(take(s(mark(z0)), cons(x1, z1))) → c5(TAKE(mark(z0), z1))
ACTIVE(take(s(z0), cons(x1, mark(z1)))) → c5(TAKE(z0, mark(z1)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(ok(eq(0, 0))) → c33(TOP(mark(true)), ACTIVE(eq(0, 0)))
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1)))
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0)))
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil)))
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0)))
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1)))
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1)))
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))
TOP(ok(x0)) → c33

S tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(x0)) → c33

K tuples:

TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(nil)) → c32(TOP(ok(nil)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))

Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c32, c33

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

Removed 5 trailing nodes:

TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(ok(x0)) → c33

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

K tuples:none
Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31

(29) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(CONS(x₁, x₂)) = [2]x₁ + [3]x₂
POL(EQ(x₁, x₂)) = [3]x₁ + [3]x₂
POL(INF(x₁)) = 0
POL(LENGTH(x₁)) = 0
POL(S(x₁)) = [3]x₁
POL(TAKE(x₁, x₂)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))

K tuples:

TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31

(31) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(CONS(x₁, x₂)) = [4]x₁ + [2]x₂
POL(EQ(x₁, x₂)) = [2]x₁ + [4]x₂
POL(INF(x₁)) = 0
POL(LENGTH(x₁)) = 0
POL(S(x₁)) = [4]x₁
POL(TAKE(x₁, x₂)) = [4]x₁ + x₂
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))

K tuples:

TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))

Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31

(33) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))

We considered the (Usable) Rules:none
And the Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(CONS(x₁, x₂)) = x₁ + x₁·x₂ + x₁²
POL(EQ(x₁, x₂)) = [3]x₁ + x₁·x₂
POL(INF(x₁)) = x₁ + x₁²
POL(LENGTH(x₁)) = 0
POL(S(x₁)) = 0
POL(TAKE(x₁, x₂)) = [2]x₂ + x₂²
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [1] + x₁

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))

K tuples:

TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))

Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31

(35) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

LENGTH(mark(z0)) → c17(LENGTH(z0))

We considered the (Usable) Rules:none
And the Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(CONS(x₁, x₂)) = x₁ + [2]x₂
POL(EQ(x₁, x₂)) = [2]x₂
POL(INF(x₁)) = [2]x₁
POL(LENGTH(x₁)) = [2]x₁
POL(S(x₁)) = [3]x₁
POL(TAKE(x₁, x₂)) = [2]x₁ + [5]x₂
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

LENGTH(ok(z0)) → c18(LENGTH(z0))

K tuples:

TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
LENGTH(mark(z0)) → c17(LENGTH(z0))

Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31

(37) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

LENGTH(ok(z0)) → c18(LENGTH(z0))

We considered the (Usable) Rules:none
And the Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(CONS(x₁, x₂)) = [2]x₂ + x₁·x₂
POL(EQ(x₁, x₂)) = [2]x₂² + [2]x₁²
POL(INF(x₁)) = 0
POL(LENGTH(x₁)) = x₁²
POL(S(x₁)) = [2]x₁²
POL(TAKE(x₁, x₂)) = 0
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [2] + x₁

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:none
K tuples:

TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))

Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31

(39) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

(7) CdtUnreachableProof (EQUIVALENT transformation)

(8) Obligation:

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

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

(19) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(20) Obligation:

(21) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(22) Obligation:

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

(24) Obligation:

(25) CdtUnreachableProof (EQUIVALENT transformation)

(26) Obligation:

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

(28) Obligation:

(29) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(30) Obligation:

(31) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(32) Obligation:

(33) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

(34) Obligation:

(35) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(36) Obligation:

(37) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

(38) Obligation:

(39) SIsEmptyProof (EQUIVALENT transformation)

(40) BOUNDS(O(1), O(1))