YES(O(1), O(n^1)) 54.36/16.91 YES(O(1), O(n^1)) 54.76/17.10 54.76/17.10 54.76/17.10 54.76/17.10 54.76/17.10 54.76/17.10 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 54.76/17.10 54.76/17.10 54.76/17.10
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
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↳2 CdtProblem
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↳3 CdtUnreachableProof (⇔)
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↳4 CdtProblem
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↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳6 CdtProblem
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↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
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↳8 CdtProblem
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↳9 CdtLeafRemovalProof (ComplexityIfPolyImplication)
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↳10 CdtProblem
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↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳12 CdtProblem
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↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
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↳14 CdtProblem
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↳15 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳16 CdtProblem
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↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳18 CdtProblem
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↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳20 CdtProblem
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↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
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↳22 CdtProblem
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↳23 CdtUnreachableProof (⇔)
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↳24 CdtProblem
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↳25 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳26 CdtProblem
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↳27 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
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↳28 CdtProblem
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↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳30 CdtProblem
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↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳32 CdtProblem
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↳33 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳34 CdtProblem
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↳35 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳36 CdtProblem
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↳37 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳38 BOUNDS(O(1), O(1))
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(0) Obligation:

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

active(eq(0, 0)) → mark(true) 54.76/17.10
active(eq(s(X), s(Y))) → mark(eq(X, Y)) 54.76/17.10
active(eq(X, Y)) → mark(false) 54.76/17.10
active(inf(X)) → mark(cons(X, inf(s(X)))) 54.76/17.10
active(take(0, X)) → mark(nil) 54.76/17.10
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L))) 54.76/17.10
active(length(nil)) → mark(0) 54.76/17.10
active(length(cons(X, L))) → mark(s(length(L))) 54.76/17.10
active(inf(X)) → inf(active(X)) 54.76/17.10
active(take(X1, X2)) → take(active(X1), X2) 54.76/17.10
active(take(X1, X2)) → take(X1, active(X2)) 54.76/17.10
active(length(X)) → length(active(X)) 54.76/17.10
inf(mark(X)) → mark(inf(X)) 54.76/17.10
take(mark(X1), X2) → mark(take(X1, X2)) 54.76/17.10
take(X1, mark(X2)) → mark(take(X1, X2)) 54.76/17.10
length(mark(X)) → mark(length(X)) 54.76/17.10
proper(eq(X1, X2)) → eq(proper(X1), proper(X2)) 54.76/17.10
proper(0) → ok(0) 54.76/17.10
proper(true) → ok(true) 54.76/17.10
proper(s(X)) → s(proper(X)) 54.76/17.10
proper(false) → ok(false) 54.76/17.10
proper(inf(X)) → inf(proper(X)) 54.76/17.10
proper(cons(any(X1), X2)) → cons(any(any(proper(X1))), any(proper(X2))) 54.76/17.10
proper(take(X1, X2)) → take(proper(X1), proper(X2)) 54.76/17.10
proper(nil) → ok(nil) 54.76/17.10
proper(length(X)) → length(proper(X)) 54.76/17.10
eq(ok(X1), ok(X2)) → ok(eq(X1, X2)) 54.76/17.10
s(ok(X)) → ok(s(X)) 54.76/17.10
inf(ok(X)) → ok(inf(X)) 54.76/17.10
cons(ok(X1), ok(X2)) → ok(cons(X1, X2)) 54.76/17.10
take(ok(X1), ok(X2)) → ok(take(X1, X2)) 54.76/17.10
length(ok(X)) → ok(length(X)) 54.76/17.10
top(mark(X)) → top(proper(X)) 54.76/17.10
top(ok(X)) → top(active(X)) 54.76/17.10
any(X) → s(X) 54.76/17.10
any(proper(X)) → any(any(any(X)))

Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
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(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 54.76/17.10
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 54.76/17.10
active(eq(z0, z1)) → mark(false) 54.76/17.10
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 54.76/17.10
active(take(0, z0)) → mark(nil) 54.76/17.10
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 54.76/17.10
active(length(nil)) → mark(0) 54.76/17.10
active(length(cons(z0, z1))) → mark(s(length(z1))) 54.76/17.10
active(inf(z0)) → inf(active(z0)) 54.76/17.10
active(take(z0, z1)) → take(active(z0), z1) 54.76/17.10
active(take(z0, z1)) → take(z0, active(z1)) 54.76/17.10
active(length(z0)) → length(active(z0)) 54.76/17.10
inf(mark(z0)) → mark(inf(z0)) 54.76/17.10
inf(ok(z0)) → ok(inf(z0)) 54.76/17.10
take(mark(z0), z1) → mark(take(z0, z1)) 54.76/17.10
take(z0, mark(z1)) → mark(take(z0, z1)) 54.76/17.10
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 54.76/17.10
length(mark(z0)) → mark(length(z0)) 54.76/17.10
length(ok(z0)) → ok(length(z0)) 54.76/17.10
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 54.76/17.10
proper(0) → ok(0) 54.76/17.10
proper(true) → ok(true) 54.76/17.10
proper(s(z0)) → s(proper(z0)) 54.76/17.10
proper(false) → ok(false) 54.76/17.10
proper(inf(z0)) → inf(proper(z0)) 54.76/17.10
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 54.76/17.10
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 54.76/17.10
proper(nil) → ok(nil) 54.76/17.10
proper(length(z0)) → length(proper(z0)) 54.76/17.10
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 54.76/17.10
s(ok(z0)) → ok(s(z0)) 54.76/17.10
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 54.76/17.10
top(mark(z0)) → top(proper(z0)) 54.76/17.10
top(ok(z0)) → top(active(z0)) 54.76/17.10
any(z0) → s(z0) 54.76/17.10
any(proper(z0)) → any(any(any(z0)))
Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 54.76/17.10
ACTIVE(inf(z0)) → c3(CONS(z0, inf(s(z0))), INF(s(z0)), S(z0)) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2)) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1)) 54.76/17.10
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 54.76/17.10
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 54.76/17.10
INF(mark(z0)) → c12(INF(z0)) 54.76/17.10
INF(ok(z0)) → c13(INF(z0)) 54.76/17.10
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 54.76/17.10
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 54.76/17.10
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 54.76/17.10
LENGTH(mark(z0)) → c17(LENGTH(z0)) 54.76/17.10
LENGTH(ok(z0)) → c18(LENGTH(z0)) 54.76/17.10
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(cons(any(z0), z1)) → c25(CONS(any(any(proper(z0))), any(proper(z1))), ANY(any(proper(z0))), ANY(proper(z0)), PROPER(z0), ANY(proper(z1)), PROPER(z1)) 54.76/17.10
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 54.76/17.10
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 54.76/17.10
S(ok(z0)) → c30(S(z0)) 54.76/17.10
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 54.76/17.10
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 54.76/17.10
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 54.76/17.10
ANY(z0) → c34(S(z0)) 54.76/17.10
ANY(proper(z0)) → c35(ANY(any(any(z0))), ANY(any(z0)), ANY(z0))
S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 54.76/17.10
ACTIVE(inf(z0)) → c3(CONS(z0, inf(s(z0))), INF(s(z0)), S(z0)) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2)) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1)) 54.76/17.10
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 54.76/17.10
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 54.76/17.10
INF(mark(z0)) → c12(INF(z0)) 54.76/17.10
INF(ok(z0)) → c13(INF(z0)) 54.76/17.10
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 54.76/17.10
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 54.76/17.10
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 54.76/17.10
LENGTH(mark(z0)) → c17(LENGTH(z0)) 54.76/17.10
LENGTH(ok(z0)) → c18(LENGTH(z0)) 54.76/17.10
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(cons(any(z0), z1)) → c25(CONS(any(any(proper(z0))), any(proper(z1))), ANY(any(proper(z0))), ANY(proper(z0)), PROPER(z0), ANY(proper(z1)), PROPER(z1)) 54.76/17.10
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 54.76/17.10
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 54.76/17.10
S(ok(z0)) → c30(S(z0)) 54.76/17.10
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 54.76/17.10
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 54.76/17.10
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 54.76/17.10
ANY(z0) → c34(S(z0)) 54.76/17.10
ANY(proper(z0)) → c35(ANY(any(any(z0))), ANY(any(z0)), ANY(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

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, c34, c35

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(3) CdtUnreachableProof (EQUIVALENT transformation)

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

PROPER(cons(any(z0), z1)) → c25(CONS(any(any(proper(z0))), any(proper(z1))), ANY(any(proper(z0))), ANY(proper(z0)), PROPER(z0), ANY(proper(z1)), PROPER(z1)) 54.76/17.10
ANY(proper(z0)) → c35(ANY(any(any(z0))), ANY(any(z0)), ANY(z0))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 54.76/17.10
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 54.76/17.10
active(eq(z0, z1)) → mark(false) 54.76/17.10
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 54.76/17.10
active(take(0, z0)) → mark(nil) 54.76/17.10
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 54.76/17.10
active(length(nil)) → mark(0) 54.76/17.10
active(length(cons(z0, z1))) → mark(s(length(z1))) 54.76/17.10
active(inf(z0)) → inf(active(z0)) 54.76/17.10
active(take(z0, z1)) → take(active(z0), z1) 54.76/17.10
active(take(z0, z1)) → take(z0, active(z1)) 54.76/17.10
active(length(z0)) → length(active(z0)) 54.76/17.10
inf(mark(z0)) → mark(inf(z0)) 54.76/17.10
inf(ok(z0)) → ok(inf(z0)) 54.76/17.10
take(mark(z0), z1) → mark(take(z0, z1)) 54.76/17.10
take(z0, mark(z1)) → mark(take(z0, z1)) 54.76/17.10
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 54.76/17.10
length(mark(z0)) → mark(length(z0)) 54.76/17.10
length(ok(z0)) → ok(length(z0)) 54.76/17.10
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 54.76/17.10
proper(0) → ok(0) 54.76/17.10
proper(true) → ok(true) 54.76/17.10
proper(s(z0)) → s(proper(z0)) 54.76/17.10
proper(false) → ok(false) 54.76/17.10
proper(inf(z0)) → inf(proper(z0)) 54.76/17.10
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 54.76/17.10
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 54.76/17.10
proper(nil) → ok(nil) 54.76/17.10
proper(length(z0)) → length(proper(z0)) 54.76/17.10
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 54.76/17.10
s(ok(z0)) → ok(s(z0)) 54.76/17.10
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 54.76/17.10
top(mark(z0)) → top(proper(z0)) 54.76/17.10
top(ok(z0)) → top(active(z0)) 54.76/17.10
any(z0) → s(z0) 54.76/17.10
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 54.76/17.10
INF(ok(z0)) → c13(INF(z0)) 54.76/17.10
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 54.76/17.10
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 54.76/17.10
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 54.76/17.10
LENGTH(mark(z0)) → c17(LENGTH(z0)) 54.76/17.10
LENGTH(ok(z0)) → c18(LENGTH(z0)) 54.76/17.10
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 54.76/17.10
S(ok(z0)) → c30(S(z0)) 54.76/17.10
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 54.76/17.10
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 54.76/17.10
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 54.76/17.10
ANY(z0) → c34(S(z0)) 54.76/17.10
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 54.76/17.10
ACTIVE(inf(z0)) → c3(CONS(z0, inf(s(z0))), INF(s(z0)), S(z0)) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2)) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1)) 54.76/17.10
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 54.76/17.10
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 54.76/17.10
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 54.76/17.10
ACTIVE(inf(z0)) → c3(CONS(z0, inf(s(z0))), INF(s(z0)), S(z0)) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2)) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1)) 54.76/17.10
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 54.76/17.10
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 54.76/17.10
INF(mark(z0)) → c12(INF(z0)) 54.76/17.10
INF(ok(z0)) → c13(INF(z0)) 54.76/17.10
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 54.76/17.10
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 54.76/17.10
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 54.76/17.10
LENGTH(mark(z0)) → c17(LENGTH(z0)) 54.76/17.10
LENGTH(ok(z0)) → c18(LENGTH(z0)) 54.76/17.10
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 54.76/17.10
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 54.76/17.10
S(ok(z0)) → c30(S(z0)) 54.76/17.10
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 54.76/17.10
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 54.76/17.10
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 54.76/17.10
ANY(z0) → c34(S(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c32, c33, c34, c1, c3, c5, c7, c8, c9, c10, c11, c19, c22, c24, c26, c28

54.76/17.10
54.76/17.10

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

Removed 2 trailing tuple parts
54.76/17.10
54.76/17.10

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 54.76/17.10
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 54.76/17.10
active(eq(z0, z1)) → mark(false) 54.76/17.10
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 54.76/17.10
active(take(0, z0)) → mark(nil) 54.76/17.10
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 54.76/17.10
active(length(nil)) → mark(0) 54.76/17.10
active(length(cons(z0, z1))) → mark(s(length(z1))) 54.76/17.10
active(inf(z0)) → inf(active(z0)) 54.76/17.10
active(take(z0, z1)) → take(active(z0), z1) 54.76/17.10
active(take(z0, z1)) → take(z0, active(z1)) 54.76/17.10
active(length(z0)) → length(active(z0)) 54.76/17.10
inf(mark(z0)) → mark(inf(z0)) 54.76/17.10
inf(ok(z0)) → ok(inf(z0)) 54.76/17.10
take(mark(z0), z1) → mark(take(z0, z1)) 54.76/17.10
take(z0, mark(z1)) → mark(take(z0, z1)) 54.76/17.10
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 54.76/17.10
length(mark(z0)) → mark(length(z0)) 54.76/17.10
length(ok(z0)) → ok(length(z0)) 54.76/17.10
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 54.76/17.10
proper(0) → ok(0) 54.76/17.10
proper(true) → ok(true) 54.76/17.10
proper(s(z0)) → s(proper(z0)) 54.76/17.10
proper(false) → ok(false) 54.76/17.10
proper(inf(z0)) → inf(proper(z0)) 54.76/17.10
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 54.76/17.10
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 54.76/17.10
proper(nil) → ok(nil) 54.76/17.10
proper(length(z0)) → length(proper(z0)) 54.76/17.10
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 54.76/17.10
s(ok(z0)) → ok(s(z0)) 54.76/17.10
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 54.76/17.10
top(mark(z0)) → top(proper(z0)) 54.76/17.10
top(ok(z0)) → top(active(z0)) 54.76/17.10
any(z0) → s(z0) 54.76/17.10
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 54.76/17.10
INF(ok(z0)) → c13(INF(z0)) 54.76/17.10
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 54.76/17.10
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 54.76/17.10
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 54.76/17.10
LENGTH(mark(z0)) → c17(LENGTH(z0)) 54.76/17.10
LENGTH(ok(z0)) → c18(LENGTH(z0)) 54.76/17.10
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 54.76/17.10
S(ok(z0)) → c30(S(z0)) 54.76/17.10
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 54.76/17.10
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 54.76/17.10
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 54.76/17.10
ANY(z0) → c34(S(z0)) 54.76/17.10
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2)) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1)) 54.76/17.10
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 54.76/17.10
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 54.76/17.10
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 54.76/17.10
ACTIVE(inf(z0)) → c3(S(z0))
S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2)) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1)) 54.76/17.10
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 54.76/17.10
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 54.76/17.10
INF(mark(z0)) → c12(INF(z0)) 54.76/17.10
INF(ok(z0)) → c13(INF(z0)) 54.76/17.10
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 54.76/17.10
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 54.76/17.10
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 54.76/17.10
LENGTH(mark(z0)) → c17(LENGTH(z0)) 54.76/17.10
LENGTH(ok(z0)) → c18(LENGTH(z0)) 54.76/17.10
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 54.76/17.10
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 54.76/17.10
S(ok(z0)) → c30(S(z0)) 54.76/17.10
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 54.76/17.10
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 54.76/17.10
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 54.76/17.10
ANY(z0) → c34(S(z0)) 54.76/17.10
ACTIVE(inf(z0)) → c3(S(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c32, c33, c34, c1, c5, c7, c8, c9, c10, c11, c19, c22, c24, c26, c28, c3

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54.76/17.10

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

Split RHS of tuples not part of any SCC
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54.76/17.10

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 54.76/17.10
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 54.76/17.10
active(eq(z0, z1)) → mark(false) 54.76/17.10
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 54.76/17.10
active(take(0, z0)) → mark(nil) 54.76/17.10
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 54.76/17.10
active(length(nil)) → mark(0) 54.76/17.10
active(length(cons(z0, z1))) → mark(s(length(z1))) 54.76/17.10
active(inf(z0)) → inf(active(z0)) 54.76/17.10
active(take(z0, z1)) → take(active(z0), z1) 54.76/17.10
active(take(z0, z1)) → take(z0, active(z1)) 54.76/17.10
active(length(z0)) → length(active(z0)) 54.76/17.10
inf(mark(z0)) → mark(inf(z0)) 54.76/17.10
inf(ok(z0)) → ok(inf(z0)) 54.76/17.10
take(mark(z0), z1) → mark(take(z0, z1)) 54.76/17.10
take(z0, mark(z1)) → mark(take(z0, z1)) 54.76/17.10
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 54.76/17.10
length(mark(z0)) → mark(length(z0)) 54.76/17.10
length(ok(z0)) → ok(length(z0)) 54.76/17.10
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 54.76/17.10
proper(0) → ok(0) 54.76/17.10
proper(true) → ok(true) 54.76/17.10
proper(s(z0)) → s(proper(z0)) 54.76/17.10
proper(false) → ok(false) 54.76/17.10
proper(inf(z0)) → inf(proper(z0)) 54.76/17.10
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 54.76/17.10
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 54.76/17.10
proper(nil) → ok(nil) 54.76/17.10
proper(length(z0)) → length(proper(z0)) 54.76/17.10
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 54.76/17.10
s(ok(z0)) → ok(s(z0)) 54.76/17.10
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 54.76/17.10
top(mark(z0)) → top(proper(z0)) 54.76/17.10
top(ok(z0)) → top(active(z0)) 54.76/17.10
any(z0) → s(z0) 54.76/17.10
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 54.76/17.10
INF(ok(z0)) → c13(INF(z0)) 54.76/17.10
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 54.76/17.10
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 54.76/17.10
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 54.76/17.10
LENGTH(mark(z0)) → c17(LENGTH(z0)) 54.76/17.10
LENGTH(ok(z0)) → c18(LENGTH(z0)) 54.76/17.10
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 54.76/17.10
S(ok(z0)) → c30(S(z0)) 54.76/17.10
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 54.76/17.10
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 54.76/17.10
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 54.76/17.10
ANY(z0) → c34(S(z0)) 54.76/17.10
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 54.76/17.10
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 54.76/17.10
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 54.76/17.10
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 54.76/17.10
ACTIVE(inf(z0)) → c3(S(z0)) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 54.76/17.10
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 54.76/17.10
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 54.76/17.10
INF(mark(z0)) → c12(INF(z0)) 54.76/17.10
INF(ok(z0)) → c13(INF(z0)) 54.76/17.10
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 54.76/17.10
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 54.76/17.10
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 54.76/17.10
LENGTH(mark(z0)) → c17(LENGTH(z0)) 54.76/17.10
LENGTH(ok(z0)) → c18(LENGTH(z0)) 54.76/17.10
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 54.76/17.10
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 54.76/17.10
S(ok(z0)) → c30(S(z0)) 54.76/17.10
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 54.76/17.10
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 54.76/17.10
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 54.76/17.10
ANY(z0) → c34(S(z0)) 54.76/17.10
ACTIVE(inf(z0)) → c3(S(z0)) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c32, c33, c34, c1, c8, c9, c10, c11, c19, c22, c24, c26, c28, c3, c

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54.76/17.10

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

ANY(z0) → c34(S(z0))
54.76/17.10
54.76/17.10

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 54.76/17.10
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 54.76/17.10
active(eq(z0, z1)) → mark(false) 54.76/17.10
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 54.76/17.10
active(take(0, z0)) → mark(nil) 54.76/17.10
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 54.76/17.10
active(length(nil)) → mark(0) 54.76/17.10
active(length(cons(z0, z1))) → mark(s(length(z1))) 54.76/17.10
active(inf(z0)) → inf(active(z0)) 54.76/17.10
active(take(z0, z1)) → take(active(z0), z1) 54.76/17.10
active(take(z0, z1)) → take(z0, active(z1)) 54.76/17.10
active(length(z0)) → length(active(z0)) 54.76/17.10
inf(mark(z0)) → mark(inf(z0)) 54.76/17.10
inf(ok(z0)) → ok(inf(z0)) 54.76/17.10
take(mark(z0), z1) → mark(take(z0, z1)) 54.76/17.10
take(z0, mark(z1)) → mark(take(z0, z1)) 54.76/17.10
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 54.76/17.10
length(mark(z0)) → mark(length(z0)) 54.76/17.10
length(ok(z0)) → ok(length(z0)) 54.76/17.10
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 54.76/17.10
proper(0) → ok(0) 54.76/17.10
proper(true) → ok(true) 54.76/17.10
proper(s(z0)) → s(proper(z0)) 54.76/17.10
proper(false) → ok(false) 54.76/17.10
proper(inf(z0)) → inf(proper(z0)) 54.76/17.10
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 54.76/17.10
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 54.76/17.10
proper(nil) → ok(nil) 54.76/17.10
proper(length(z0)) → length(proper(z0)) 54.76/17.10
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 54.76/17.10
s(ok(z0)) → ok(s(z0)) 54.76/17.10
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 54.76/17.10
top(mark(z0)) → top(proper(z0)) 54.76/17.10
top(ok(z0)) → top(active(z0)) 54.76/17.10
any(z0) → s(z0) 54.76/17.10
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 54.76/17.10
INF(ok(z0)) → c13(INF(z0)) 54.76/17.10
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 54.76/17.10
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 54.76/17.10
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 54.76/17.10
LENGTH(mark(z0)) → c17(LENGTH(z0)) 54.76/17.10
LENGTH(ok(z0)) → c18(LENGTH(z0)) 54.76/17.10
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 54.76/17.10
S(ok(z0)) → c30(S(z0)) 54.76/17.10
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 54.76/17.10
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 54.76/17.10
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 54.76/17.10
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 54.76/17.10
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 54.76/17.10
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 54.76/17.10
ACTIVE(inf(z0)) → c3(S(z0)) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 54.76/17.10
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 54.76/17.10
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 54.76/17.10
INF(mark(z0)) → c12(INF(z0)) 54.76/17.10
INF(ok(z0)) → c13(INF(z0)) 54.76/17.10
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 54.76/17.10
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 54.76/17.10
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 54.76/17.10
LENGTH(mark(z0)) → c17(LENGTH(z0)) 54.76/17.10
LENGTH(ok(z0)) → c18(LENGTH(z0)) 54.76/17.10
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 54.76/17.10
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 54.76/17.10
S(ok(z0)) → c30(S(z0)) 54.76/17.10
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 54.76/17.10
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 54.76/17.10
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(inf(z0)) → c3(S(z0)) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
K tuples:none
Defined Rule Symbols:

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

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, c8, c9, c10, c11, c19, c22, c24, c26, c28, c3, c

54.76/17.10
54.76/17.10

(11) 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.

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
We considered the (Usable) Rules:

length(mark(z0)) → mark(length(z0)) 54.76/17.10
length(ok(z0)) → ok(length(z0)) 54.76/17.10
take(mark(z0), z1) → mark(take(z0, z1)) 54.76/17.10
take(z0, mark(z1)) → mark(take(z0, z1)) 54.76/17.10
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 54.76/17.10
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 54.76/17.10
proper(0) → ok(0) 54.76/17.10
proper(true) → ok(true) 54.76/17.10
proper(s(z0)) → s(proper(z0)) 54.76/17.10
proper(false) → ok(false) 54.76/17.10
proper(inf(z0)) → inf(proper(z0)) 54.76/17.10
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 54.76/17.10
proper(nil) → ok(nil) 54.76/17.10
proper(length(z0)) → length(proper(z0)) 54.76/17.10
inf(mark(z0)) → mark(inf(z0)) 54.76/17.10
inf(ok(z0)) → ok(inf(z0)) 54.76/17.10
s(ok(z0)) → ok(s(z0)) 54.76/17.10
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 54.76/17.10
active(eq(0, 0)) → mark(true) 54.76/17.10
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 54.76/17.10
active(eq(z0, z1)) → mark(false) 54.76/17.10
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 54.76/17.10
active(take(0, z0)) → mark(nil) 54.76/17.10
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 54.76/17.10
active(length(nil)) → mark(0) 54.76/17.10
active(length(cons(z0, z1))) → mark(s(length(z1))) 54.76/17.10
active(inf(z0)) → inf(active(z0)) 54.76/17.10
active(take(z0, z1)) → take(active(z0), z1) 54.76/17.10
active(take(z0, z1)) → take(z0, active(z1)) 54.76/17.10
active(length(z0)) → length(active(z0)) 54.76/17.10
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
And the Tuples:

INF(mark(z0)) → c12(INF(z0)) 54.76/17.10
INF(ok(z0)) → c13(INF(z0)) 54.76/17.10
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 54.76/17.10
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 54.76/17.10
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 54.76/17.10
LENGTH(mark(z0)) → c17(LENGTH(z0)) 54.76/17.10
LENGTH(ok(z0)) → c18(LENGTH(z0)) 54.76/17.10
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 54.76/17.10
S(ok(z0)) → c30(S(z0)) 54.76/17.10
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 54.76/17.10
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 54.76/17.10
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 54.76/17.10
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 54.76/17.10
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 54.76/17.10
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 54.76/17.10
ACTIVE(inf(z0)) → c3(S(z0)) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 54.76/17.10

POL(0) = 0    54.76/17.10
POL(ACTIVE(x1)) = 0    54.76/17.10
POL(CONS(x1, x2)) = x2    54.76/17.10
POL(EQ(x1, x2)) = 0    54.76/17.10
POL(INF(x1)) = 0    54.76/17.10
POL(LENGTH(x1)) = 0    54.76/17.10
POL(PROPER(x1)) = 0    54.76/17.10
POL(S(x1)) = 0    54.76/17.10
POL(TAKE(x1, x2)) = 0    54.76/17.10
POL(TOP(x1)) = 0    54.76/17.10
POL(active(x1)) = 0    54.76/17.10
POL(c(x1)) = x1    54.76/17.10
POL(c1(x1)) = x1    54.76/17.10
POL(c10(x1, x2)) = x1 + x2    54.76/17.10
POL(c11(x1, x2)) = x1 + x2    54.76/17.10
POL(c12(x1)) = x1    54.76/17.10
POL(c13(x1)) = x1    54.76/17.10
POL(c14(x1)) = x1    54.76/17.10
POL(c15(x1)) = x1    54.76/17.10
POL(c16(x1)) = x1    54.76/17.10
POL(c17(x1)) = x1    54.76/17.10
POL(c18(x1)) = x1    54.76/17.10
POL(c19(x1, x2, x3)) = x1 + x2 + x3    54.76/17.10
POL(c22(x1, x2)) = x1 + x2    54.76/17.10
POL(c24(x1, x2)) = x1 + x2    54.76/17.10
POL(c26(x1, x2, x3)) = x1 + x2 + x3    54.76/17.10
POL(c28(x1, x2)) = x1 + x2    54.76/17.10
POL(c29(x1)) = x1    54.76/17.10
POL(c3(x1)) = x1    54.76/17.10
POL(c30(x1)) = x1    54.76/17.10
POL(c31(x1)) = x1    54.76/17.10
POL(c32(x1, x2)) = x1 + x2    54.76/17.10
POL(c33(x1, x2)) = x1 + x2    54.76/17.10
POL(c8(x1, x2)) = x1 + x2    54.76/17.10
POL(c9(x1, x2)) = x1 + x2    54.76/17.10
POL(cons(x1, x2)) = 0    54.76/17.10
POL(eq(x1, x2)) = 0    54.76/17.10
POL(false) = 0    54.76/17.10
POL(inf(x1)) = 0    54.76/17.10
POL(length(x1)) = 0    54.76/17.10
POL(mark(x1)) = 0    54.76/17.10
POL(nil) = 0    54.76/17.10
POL(ok(x1)) = [4] + x1    54.76/17.10
POL(proper(x1)) = 0    54.76/17.10
POL(s(x1)) = 0    54.76/17.10
POL(take(x1, x2)) = 0    54.76/17.10
POL(true) = 0   
54.76/17.10
54.76/17.10

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 54.76/17.10
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 54.76/17.10
active(eq(z0, z1)) → mark(false) 54.76/17.10
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 54.76/17.10
active(take(0, z0)) → mark(nil) 54.76/17.10
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 54.76/17.10
active(length(nil)) → mark(0) 54.76/17.10
active(length(cons(z0, z1))) → mark(s(length(z1))) 54.76/17.10
active(inf(z0)) → inf(active(z0)) 54.76/17.10
active(take(z0, z1)) → take(active(z0), z1) 54.76/17.10
active(take(z0, z1)) → take(z0, active(z1)) 54.76/17.10
active(length(z0)) → length(active(z0)) 54.76/17.10
inf(mark(z0)) → mark(inf(z0)) 54.76/17.10
inf(ok(z0)) → ok(inf(z0)) 54.76/17.10
take(mark(z0), z1) → mark(take(z0, z1)) 54.76/17.10
take(z0, mark(z1)) → mark(take(z0, z1)) 54.76/17.10
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 54.76/17.10
length(mark(z0)) → mark(length(z0)) 54.76/17.10
length(ok(z0)) → ok(length(z0)) 54.76/17.10
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 54.76/17.10
proper(0) → ok(0) 54.76/17.10
proper(true) → ok(true) 54.76/17.10
proper(s(z0)) → s(proper(z0)) 54.76/17.10
proper(false) → ok(false) 54.76/17.10
proper(inf(z0)) → inf(proper(z0)) 54.76/17.10
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 54.76/17.10
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 54.76/17.10
proper(nil) → ok(nil) 54.76/17.10
proper(length(z0)) → length(proper(z0)) 54.76/17.10
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 54.76/17.10
s(ok(z0)) → ok(s(z0)) 54.76/17.10
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 54.76/17.10
top(mark(z0)) → top(proper(z0)) 54.76/17.10
top(ok(z0)) → top(active(z0)) 54.76/17.10
any(z0) → s(z0) 54.76/17.10
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 54.76/17.10
INF(ok(z0)) → c13(INF(z0)) 54.76/17.10
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 54.76/17.10
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 54.76/17.10
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 54.76/17.10
LENGTH(mark(z0)) → c17(LENGTH(z0)) 54.76/17.10
LENGTH(ok(z0)) → c18(LENGTH(z0)) 54.76/17.10
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 54.76/17.10
S(ok(z0)) → c30(S(z0)) 54.76/17.10
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 54.76/17.10
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 54.76/17.10
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 54.76/17.10
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 54.76/17.10
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 54.76/17.10
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 54.76/17.10
ACTIVE(inf(z0)) → c3(S(z0)) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 54.76/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 54.76/17.10
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 54.76/17.10
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 54.76/17.10
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 54.76/17.10
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 54.76/17.10
INF(mark(z0)) → c12(INF(z0)) 54.76/17.10
INF(ok(z0)) → c13(INF(z0)) 54.76/17.10
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 54.76/17.10
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 54.76/17.10
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 54.76/17.10
LENGTH(mark(z0)) → c17(LENGTH(z0)) 54.76/17.10
LENGTH(ok(z0)) → c18(LENGTH(z0)) 54.76/17.10
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 54.76/17.10
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 54.76/17.10
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 55.15/17.10
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.10
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 55.15/17.10
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.10
S(ok(z0)) → c30(S(z0)) 55.15/17.10
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 55.15/17.10
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 55.15/17.10
ACTIVE(inf(z0)) → c3(S(z0)) 55.15/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 55.15/17.10
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 55.15/17.10
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 55.15/17.10
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
K tuples:

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
Defined Rule Symbols:

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

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, c8, c9, c10, c11, c19, c22, c24, c26, c28, c3, c

55.15/17.10
55.15/17.10

(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))) 55.15/17.10
TOP(mark(0)) → c32(TOP(ok(0)), PROPER(0)) 55.15/17.10
TOP(mark(true)) → c32(TOP(ok(true)), PROPER(true)) 55.15/17.11
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 55.15/17.11
TOP(mark(false)) → c32(TOP(ok(false)), PROPER(false)) 55.15/17.11
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 55.15/17.11
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 55.15/17.11
TOP(mark(nil)) → c32(TOP(ok(nil)), PROPER(nil)) 55.15/17.11
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
55.15/17.11
55.15/17.11

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 55.15/17.11
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 55.15/17.11
active(eq(z0, z1)) → mark(false) 55.15/17.11
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 55.15/17.11
active(take(0, z0)) → mark(nil) 55.15/17.11
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 55.15/17.11
active(length(nil)) → mark(0) 55.15/17.11
active(length(cons(z0, z1))) → mark(s(length(z1))) 55.15/17.11
active(inf(z0)) → inf(active(z0)) 55.15/17.11
active(take(z0, z1)) → take(active(z0), z1) 55.15/17.11
active(take(z0, z1)) → take(z0, active(z1)) 55.15/17.11
active(length(z0)) → length(active(z0)) 55.15/17.11
inf(mark(z0)) → mark(inf(z0)) 55.15/17.11
inf(ok(z0)) → ok(inf(z0)) 55.15/17.11
take(mark(z0), z1) → mark(take(z0, z1)) 55.15/17.11
take(z0, mark(z1)) → mark(take(z0, z1)) 55.15/17.11
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 55.15/17.11
length(mark(z0)) → mark(length(z0)) 55.15/17.11
length(ok(z0)) → ok(length(z0)) 55.15/17.11
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 55.15/17.11
proper(0) → ok(0) 55.15/17.11
proper(true) → ok(true) 55.15/17.11
proper(s(z0)) → s(proper(z0)) 55.15/17.11
proper(false) → ok(false) 55.15/17.11
proper(inf(z0)) → inf(proper(z0)) 55.15/17.11
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 55.15/17.11
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 55.15/17.11
proper(nil) → ok(nil) 55.15/17.11
proper(length(z0)) → length(proper(z0)) 55.15/17.11
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 55.15/17.11
s(ok(z0)) → ok(s(z0)) 55.15/17.11
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 55.15/17.11
top(mark(z0)) → top(proper(z0)) 55.15/17.11
top(ok(z0)) → top(active(z0)) 55.15/17.11
any(z0) → s(z0) 55.15/17.11
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.11
INF(ok(z0)) → c13(INF(z0)) 55.15/17.11
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.11
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.11
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.11
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.11
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.11
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.11
S(ok(z0)) → c30(S(z0)) 55.15/17.11
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.11
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 55.15/17.11
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 55.15/17.11
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 55.15/17.11
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 55.15/17.11
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 55.15/17.11
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 55.15/17.11
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.11
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 55.15/17.11
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 55.15/17.11
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.11
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 55.15/17.11
ACTIVE(inf(z0)) → c3(S(z0)) 55.15/17.11
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 55.15/17.11
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 55.15/17.11
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 55.15/17.11
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 55.15/17.11
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 55.15/17.11
TOP(mark(0)) → c32(TOP(ok(0)), PROPER(0)) 55.15/17.11
TOP(mark(true)) → c32(TOP(ok(true)), PROPER(true)) 55.15/17.11
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 55.15/17.11
TOP(mark(false)) → c32(TOP(ok(false)), PROPER(false)) 55.15/17.11
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 55.15/17.11
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 55.15/17.11
TOP(mark(nil)) → c32(TOP(ok(nil)), PROPER(nil)) 55.15/17.11
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 55.15/17.11
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 55.15/17.11
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 55.15/17.11
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 55.15/17.11
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 55.15/17.11
INF(mark(z0)) → c12(INF(z0)) 55.15/17.11
INF(ok(z0)) → c13(INF(z0)) 55.15/17.11
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.11
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.11
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.11
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.11
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.11
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.11
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 55.15/17.11
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 55.15/17.11
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.11
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 55.15/17.11
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.11
S(ok(z0)) → c30(S(z0)) 55.15/17.11
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 55.15/17.11
ACTIVE(inf(z0)) → c3(S(z0)) 55.15/17.11
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 55.15/17.11
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 55.15/17.11
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 55.15/17.11
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 55.15/17.11
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 55.15/17.11
TOP(mark(0)) → c32(TOP(ok(0)), PROPER(0)) 55.15/17.11
TOP(mark(true)) → c32(TOP(ok(true)), PROPER(true)) 55.15/17.11
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 55.15/17.11
TOP(mark(false)) → c32(TOP(ok(false)), PROPER(false)) 55.15/17.11
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 55.15/17.11
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 55.15/17.11
TOP(mark(nil)) → c32(TOP(ok(nil)), PROPER(nil)) 55.15/17.11
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
K tuples:

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
Defined Rule Symbols:

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

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, c8, c9, c10, c11, c19, c22, c24, c26, c28, c3, c, c32

55.15/17.11
55.15/17.11

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

Removed 4 trailing tuple parts
55.15/17.11
55.15/17.11

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 55.15/17.11
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 55.15/17.11
active(eq(z0, z1)) → mark(false) 55.15/17.11
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 55.15/17.11
active(take(0, z0)) → mark(nil) 55.15/17.11
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 55.15/17.11
active(length(nil)) → mark(0) 55.15/17.11
active(length(cons(z0, z1))) → mark(s(length(z1))) 55.15/17.11
active(inf(z0)) → inf(active(z0)) 55.15/17.11
active(take(z0, z1)) → take(active(z0), z1) 55.15/17.11
active(take(z0, z1)) → take(z0, active(z1)) 55.15/17.11
active(length(z0)) → length(active(z0)) 55.15/17.11
inf(mark(z0)) → mark(inf(z0)) 55.15/17.11
inf(ok(z0)) → ok(inf(z0)) 55.15/17.11
take(mark(z0), z1) → mark(take(z0, z1)) 55.15/17.11
take(z0, mark(z1)) → mark(take(z0, z1)) 55.15/17.11
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 55.15/17.11
length(mark(z0)) → mark(length(z0)) 55.15/17.11
length(ok(z0)) → ok(length(z0)) 55.15/17.11
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 55.15/17.11
proper(0) → ok(0) 55.15/17.11
proper(true) → ok(true) 55.15/17.11
proper(s(z0)) → s(proper(z0)) 55.15/17.11
proper(false) → ok(false) 55.15/17.11
proper(inf(z0)) → inf(proper(z0)) 55.15/17.11
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 55.15/17.11
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 55.15/17.11
proper(nil) → ok(nil) 55.15/17.11
proper(length(z0)) → length(proper(z0)) 55.15/17.11
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 55.15/17.11
s(ok(z0)) → ok(s(z0)) 55.15/17.11
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 55.15/17.11
top(mark(z0)) → top(proper(z0)) 55.15/17.11
top(ok(z0)) → top(active(z0)) 55.15/17.11
any(z0) → s(z0) 55.15/17.11
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.11
INF(ok(z0)) → c13(INF(z0)) 55.15/17.11
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.11
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.11
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.11
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.11
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.11
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.11
S(ok(z0)) → c30(S(z0)) 55.15/17.11
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.11
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 55.15/17.11
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 55.15/17.11
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 55.15/17.11
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 55.15/17.11
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 55.15/17.11
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 55.15/17.11
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.11
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 55.15/17.11
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 55.15/17.11
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.11
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 55.15/17.11
ACTIVE(inf(z0)) → c3(S(z0)) 55.15/17.11
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 55.15/17.11
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 55.15/17.11
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 55.15/17.11
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 55.15/17.11
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 55.15/17.11
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 55.15/17.11
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 55.15/17.11
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 55.15/17.11
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 55.15/17.11
TOP(mark(0)) → c32(TOP(ok(0))) 55.15/17.11
TOP(mark(true)) → c32(TOP(ok(true))) 55.15/17.11
TOP(mark(false)) → c32(TOP(ok(false))) 55.15/17.11
TOP(mark(nil)) → c32(TOP(ok(nil)))
S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 55.15/17.11
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 55.15/17.11
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 55.15/17.11
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 55.15/17.11
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 55.15/17.11
INF(mark(z0)) → c12(INF(z0)) 55.15/17.11
INF(ok(z0)) → c13(INF(z0)) 55.15/17.11
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.11
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.11
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.11
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.11
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.11
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.11
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 55.15/17.11
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 55.15/17.11
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.11
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 55.15/17.11
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.11
S(ok(z0)) → c30(S(z0)) 55.15/17.11
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 55.15/17.12
ACTIVE(inf(z0)) → c3(S(z0)) 55.15/17.12
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 55.15/17.12
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 55.15/17.12
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 55.15/17.12
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 55.15/17.12
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 55.15/17.12
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 55.15/17.12
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 55.15/17.12
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 55.15/17.12
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 55.15/17.12
TOP(mark(0)) → c32(TOP(ok(0))) 55.15/17.12
TOP(mark(true)) → c32(TOP(ok(true))) 55.15/17.12
TOP(mark(false)) → c32(TOP(ok(false))) 55.15/17.12
TOP(mark(nil)) → c32(TOP(ok(nil)))
K tuples:

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
Defined Rule Symbols:

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

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, c8, c9, c10, c11, c19, c22, c24, c26, c28, c3, c, c32, c32

55.15/17.12
55.15/17.12

(17) 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))) 55.15/17.12
TOP(mark(0)) → c32(TOP(ok(0))) 55.15/17.12
TOP(mark(true)) → c32(TOP(ok(true))) 55.15/17.12
TOP(mark(false)) → c32(TOP(ok(false)))
We considered the (Usable) Rules:

proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 55.15/17.12
proper(0) → ok(0) 55.15/17.12
proper(true) → ok(true) 55.15/17.12
proper(s(z0)) → s(proper(z0)) 55.15/17.12
proper(false) → ok(false) 55.15/17.12
proper(inf(z0)) → inf(proper(z0)) 55.15/17.12
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 55.15/17.12
proper(nil) → ok(nil) 55.15/17.12
proper(length(z0)) → length(proper(z0)) 55.15/17.12
length(mark(z0)) → mark(length(z0)) 55.15/17.12
length(ok(z0)) → ok(length(z0)) 55.15/17.12
take(mark(z0), z1) → mark(take(z0, z1)) 55.15/17.12
take(z0, mark(z1)) → mark(take(z0, z1)) 55.15/17.12
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 55.15/17.12
inf(mark(z0)) → mark(inf(z0)) 55.15/17.12
inf(ok(z0)) → ok(inf(z0)) 55.15/17.12
s(ok(z0)) → ok(s(z0)) 55.15/17.12
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 55.15/17.12
active(eq(0, 0)) → mark(true) 55.15/17.12
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 55.15/17.12
active(eq(z0, z1)) → mark(false) 55.15/17.12
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 55.15/17.12
active(take(0, z0)) → mark(nil) 55.15/17.12
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 55.15/17.12
active(length(nil)) → mark(0) 55.15/17.12
active(length(cons(z0, z1))) → mark(s(length(z1))) 55.15/17.12
active(inf(z0)) → inf(active(z0)) 55.15/17.12
active(take(z0, z1)) → take(active(z0), z1) 55.15/17.12
active(take(z0, z1)) → take(z0, active(z1)) 55.15/17.12
active(length(z0)) → length(active(z0)) 55.15/17.12
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
And the Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.12
INF(ok(z0)) → c13(INF(z0)) 55.15/17.12
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.12
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.12
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.12
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.12
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.12
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.12
S(ok(z0)) → c30(S(z0)) 55.15/17.12
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.12
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 55.15/17.12
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 55.15/17.12
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 55.15/17.12
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 55.15/17.12
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 55.15/17.12
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 55.15/17.12
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.12
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 55.15/17.12
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 55.15/17.12
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.12
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 55.15/17.12
ACTIVE(inf(z0)) → c3(S(z0)) 55.15/17.12
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 55.15/17.12
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 55.15/17.12
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 55.15/17.12
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 55.15/17.12
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 55.15/17.12
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 55.15/17.12
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 55.15/17.12
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 55.15/17.12
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 55.15/17.12
TOP(mark(0)) → c32(TOP(ok(0))) 55.15/17.12
TOP(mark(true)) → c32(TOP(ok(true))) 55.15/17.12
TOP(mark(false)) → c32(TOP(ok(false))) 55.15/17.12
TOP(mark(nil)) → c32(TOP(ok(nil)))
The order we found is given by the following interpretation:
Polynomial interpretation : 55.15/17.12

POL(0) = 0    55.15/17.12
POL(ACTIVE(x1)) = 0    55.15/17.12
POL(CONS(x1, x2)) = 0    55.15/17.12
POL(EQ(x1, x2)) = 0    55.15/17.12
POL(INF(x1)) = 0    55.15/17.12
POL(LENGTH(x1)) = 0    55.15/17.12
POL(PROPER(x1)) = 0    55.15/17.12
POL(S(x1)) = 0    55.15/17.12
POL(TAKE(x1, x2)) = 0    55.15/17.12
POL(TOP(x1)) = x1    55.15/17.12
POL(active(x1)) = x1    55.15/17.12
POL(c(x1)) = x1    55.15/17.12
POL(c1(x1)) = x1    55.15/17.12
POL(c10(x1, x2)) = x1 + x2    55.15/17.12
POL(c11(x1, x2)) = x1 + x2    55.15/17.12
POL(c12(x1)) = x1    55.15/17.12
POL(c13(x1)) = x1    55.15/17.12
POL(c14(x1)) = x1    55.15/17.12
POL(c15(x1)) = x1    55.15/17.12
POL(c16(x1)) = x1    55.15/17.12
POL(c17(x1)) = x1    55.15/17.12
POL(c18(x1)) = x1    55.15/17.12
POL(c19(x1, x2, x3)) = x1 + x2 + x3    55.15/17.12
POL(c22(x1, x2)) = x1 + x2    55.15/17.12
POL(c24(x1, x2)) = x1 + x2    55.15/17.12
POL(c26(x1, x2, x3)) = x1 + x2 + x3    55.15/17.12
POL(c28(x1, x2)) = x1 + x2    55.15/17.12
POL(c29(x1)) = x1    55.15/17.12
POL(c3(x1)) = x1    55.15/17.12
POL(c30(x1)) = x1    55.15/17.12
POL(c31(x1)) = x1    55.15/17.12
POL(c32(x1)) = x1    55.15/17.12
POL(c32(x1, x2)) = x1 + x2    55.15/17.12
POL(c33(x1, x2)) = x1 + x2    55.15/17.12
POL(c8(x1, x2)) = x1 + x2    55.15/17.12
POL(c9(x1, x2)) = x1 + x2    55.15/17.12
POL(cons(x1, x2)) = 0    55.15/17.12
POL(eq(x1, x2)) = [1]    55.15/17.12
POL(false) = 0    55.15/17.12
POL(inf(x1)) = [1]    55.15/17.12
POL(length(x1)) = [1]    55.15/17.12
POL(mark(x1)) = [1]    55.15/17.12
POL(nil) = [1]    55.15/17.12
POL(ok(x1)) = x1    55.15/17.12
POL(proper(x1)) = 0    55.15/17.12
POL(s(x1)) = 0    55.15/17.12
POL(take(x1, x2)) = [1]    55.15/17.12
POL(true) = 0   
55.15/17.12
55.15/17.12

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 55.15/17.12
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 55.15/17.12
active(eq(z0, z1)) → mark(false) 55.15/17.12
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 55.15/17.12
active(take(0, z0)) → mark(nil) 55.15/17.12
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 55.15/17.12
active(length(nil)) → mark(0) 55.15/17.12
active(length(cons(z0, z1))) → mark(s(length(z1))) 55.15/17.12
active(inf(z0)) → inf(active(z0)) 55.15/17.12
active(take(z0, z1)) → take(active(z0), z1) 55.15/17.12
active(take(z0, z1)) → take(z0, active(z1)) 55.15/17.12
active(length(z0)) → length(active(z0)) 55.15/17.12
inf(mark(z0)) → mark(inf(z0)) 55.15/17.12
inf(ok(z0)) → ok(inf(z0)) 55.15/17.12
take(mark(z0), z1) → mark(take(z0, z1)) 55.15/17.12
take(z0, mark(z1)) → mark(take(z0, z1)) 55.15/17.12
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 55.15/17.12
length(mark(z0)) → mark(length(z0)) 55.15/17.12
length(ok(z0)) → ok(length(z0)) 55.15/17.12
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 55.15/17.12
proper(0) → ok(0) 55.15/17.12
proper(true) → ok(true) 55.15/17.12
proper(s(z0)) → s(proper(z0)) 55.15/17.12
proper(false) → ok(false) 55.15/17.12
proper(inf(z0)) → inf(proper(z0)) 55.15/17.12
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 55.15/17.12
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 55.15/17.12
proper(nil) → ok(nil) 55.15/17.12
proper(length(z0)) → length(proper(z0)) 55.15/17.12
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 55.15/17.12
s(ok(z0)) → ok(s(z0)) 55.15/17.12
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 55.15/17.12
top(mark(z0)) → top(proper(z0)) 55.15/17.12
top(ok(z0)) → top(active(z0)) 55.15/17.12
any(z0) → s(z0) 55.15/17.12
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.12
INF(ok(z0)) → c13(INF(z0)) 55.15/17.12
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.12
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.12
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.12
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.12
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.12
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.12
S(ok(z0)) → c30(S(z0)) 55.15/17.12
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.12
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 55.15/17.12
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 55.15/17.12
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 55.15/17.12
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 55.15/17.12
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 55.15/17.12
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 55.15/17.12
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.12
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 55.15/17.12
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 55.15/17.12
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.12
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 55.15/17.12
ACTIVE(inf(z0)) → c3(S(z0)) 55.15/17.12
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 55.15/17.12
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 55.15/17.12
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 55.15/17.12
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 55.15/17.12
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 55.15/17.12
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 55.15/17.12
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 55.15/17.12
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 55.15/17.12
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 55.15/17.12
TOP(mark(0)) → c32(TOP(ok(0))) 55.15/17.12
TOP(mark(true)) → c32(TOP(ok(true))) 55.15/17.12
TOP(mark(false)) → c32(TOP(ok(false))) 55.15/17.12
TOP(mark(nil)) → c32(TOP(ok(nil)))
S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 55.15/17.12
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 55.15/17.12
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 55.15/17.12
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 55.15/17.12
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 55.15/17.12
INF(mark(z0)) → c12(INF(z0)) 55.15/17.12
INF(ok(z0)) → c13(INF(z0)) 55.15/17.12
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.12
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.12
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.12
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.12
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.12
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.12
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 55.15/17.12
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 55.15/17.12
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.12
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 55.15/17.12
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.12
S(ok(z0)) → c30(S(z0)) 55.15/17.12
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 55.15/17.12
ACTIVE(inf(z0)) → c3(S(z0)) 55.15/17.12
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 55.15/17.12
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 55.15/17.12
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 55.15/17.12
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 55.15/17.12
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 55.15/17.12
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 55.15/17.12
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 55.15/17.12
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 55.15/17.12
TOP(mark(nil)) → c32(TOP(ok(nil)))
K tuples:

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.12
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 55.15/17.12
TOP(mark(0)) → c32(TOP(ok(0))) 55.15/17.12
TOP(mark(true)) → c32(TOP(ok(true))) 55.15/17.12
TOP(mark(false)) → c32(TOP(ok(false)))
Defined Rule Symbols:

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

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, c8, c9, c10, c11, c19, c22, c24, c26, c28, c3, c, c32, c32

55.15/17.12
55.15/17.12

(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(nil)) → c32(TOP(ok(nil)))
We considered the (Usable) Rules:

proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 55.15/17.12
proper(0) → ok(0) 55.15/17.12
proper(true) → ok(true) 55.15/17.12
proper(s(z0)) → s(proper(z0)) 55.15/17.12
proper(false) → ok(false) 55.15/17.12
proper(inf(z0)) → inf(proper(z0)) 55.15/17.12
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 55.15/17.12
proper(nil) → ok(nil) 55.15/17.12
proper(length(z0)) → length(proper(z0)) 55.15/17.12
length(mark(z0)) → mark(length(z0)) 55.15/17.12
length(ok(z0)) → ok(length(z0)) 55.15/17.12
take(mark(z0), z1) → mark(take(z0, z1)) 55.15/17.12
take(z0, mark(z1)) → mark(take(z0, z1)) 55.15/17.12
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 55.15/17.12
inf(mark(z0)) → mark(inf(z0)) 55.15/17.12
inf(ok(z0)) → ok(inf(z0)) 55.15/17.12
s(ok(z0)) → ok(s(z0)) 55.15/17.12
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 55.15/17.12
active(eq(0, 0)) → mark(true) 55.15/17.12
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 55.15/17.12
active(eq(z0, z1)) → mark(false) 55.15/17.12
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 55.15/17.12
active(take(0, z0)) → mark(nil) 55.15/17.12
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 55.15/17.12
active(length(nil)) → mark(0) 55.15/17.12
active(length(cons(z0, z1))) → mark(s(length(z1))) 55.15/17.12
active(inf(z0)) → inf(active(z0)) 55.15/17.12
active(take(z0, z1)) → take(active(z0), z1) 55.15/17.12
active(take(z0, z1)) → take(z0, active(z1)) 55.15/17.12
active(length(z0)) → length(active(z0)) 55.15/17.12
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
And the Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.12
INF(ok(z0)) → c13(INF(z0)) 55.15/17.12
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.12
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.12
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.12
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.12
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.12
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.12
S(ok(z0)) → c30(S(z0)) 55.15/17.12
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.12
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 55.15/17.12
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 55.15/17.12
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 55.15/17.12
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 55.15/17.12
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 55.15/17.12
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 55.15/17.12
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.12
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 55.15/17.12
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 55.15/17.12
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.12
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 55.15/17.12
ACTIVE(inf(z0)) → c3(S(z0)) 55.15/17.12
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 55.15/17.12
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 55.15/17.12
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 55.15/17.12
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 55.15/17.12
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 55.15/17.12
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 55.15/17.12
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 55.15/17.12
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 55.15/17.12
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 55.15/17.12
TOP(mark(0)) → c32(TOP(ok(0))) 55.15/17.12
TOP(mark(true)) → c32(TOP(ok(true))) 55.15/17.12
TOP(mark(false)) → c32(TOP(ok(false))) 55.15/17.12
TOP(mark(nil)) → c32(TOP(ok(nil)))
The order we found is given by the following interpretation:
Polynomial interpretation : 55.15/17.12

POL(0) = 0    55.15/17.12
POL(ACTIVE(x1)) = 0    55.15/17.12
POL(CONS(x1, x2)) = 0    55.15/17.12
POL(EQ(x1, x2)) = 0    55.15/17.12
POL(INF(x1)) = 0    55.15/17.12
POL(LENGTH(x1)) = 0    55.15/17.12
POL(PROPER(x1)) = 0    55.15/17.12
POL(S(x1)) = 0    55.15/17.12
POL(TAKE(x1, x2)) = 0    55.15/17.12
POL(TOP(x1)) = [4]x1    55.15/17.12
POL(active(x1)) = x1    55.15/17.12
POL(c(x1)) = x1    55.15/17.12
POL(c1(x1)) = x1    55.15/17.12
POL(c10(x1, x2)) = x1 + x2    55.15/17.12
POL(c11(x1, x2)) = x1 + x2    55.15/17.12
POL(c12(x1)) = x1    55.15/17.12
POL(c13(x1)) = x1    55.15/17.12
POL(c14(x1)) = x1    55.15/17.12
POL(c15(x1)) = x1    55.15/17.12
POL(c16(x1)) = x1    55.15/17.12
POL(c17(x1)) = x1    55.15/17.12
POL(c18(x1)) = x1    55.15/17.12
POL(c19(x1, x2, x3)) = x1 + x2 + x3    55.15/17.12
POL(c22(x1, x2)) = x1 + x2    55.15/17.12
POL(c24(x1, x2)) = x1 + x2    55.15/17.12
POL(c26(x1, x2, x3)) = x1 + x2 + x3    55.15/17.12
POL(c28(x1, x2)) = x1 + x2    55.15/17.12
POL(c29(x1)) = x1    55.15/17.12
POL(c3(x1)) = x1    55.15/17.12
POL(c30(x1)) = x1    55.15/17.12
POL(c31(x1)) = x1    55.15/17.12
POL(c32(x1)) = x1    55.15/17.12
POL(c32(x1, x2)) = x1 + x2    55.15/17.12
POL(c33(x1, x2)) = x1 + x2    55.15/17.12
POL(c8(x1, x2)) = x1 + x2    55.15/17.12
POL(c9(x1, x2)) = x1 + x2    55.15/17.12
POL(cons(x1, x2)) = 0    55.15/17.12
POL(eq(x1, x2)) = [4]    55.15/17.12
POL(false) = 0    55.15/17.12
POL(inf(x1)) = [4]    55.15/17.12
POL(length(x1)) = [4]    55.15/17.12
POL(mark(x1)) = [4]    55.15/17.12
POL(nil) = 0    55.15/17.12
POL(ok(x1)) = x1    55.15/17.12
POL(proper(x1)) = 0    55.15/17.12
POL(s(x1)) = [4]    55.15/17.12
POL(take(x1, x2)) = [4]    55.15/17.12
POL(true) = [4]   
55.15/17.12
55.15/17.12

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 55.15/17.12
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 55.15/17.12
active(eq(z0, z1)) → mark(false) 55.15/17.12
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 55.15/17.12
active(take(0, z0)) → mark(nil) 55.15/17.12
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 55.15/17.12
active(length(nil)) → mark(0) 55.15/17.12
active(length(cons(z0, z1))) → mark(s(length(z1))) 55.15/17.12
active(inf(z0)) → inf(active(z0)) 55.15/17.12
active(take(z0, z1)) → take(active(z0), z1) 55.15/17.12
active(take(z0, z1)) → take(z0, active(z1)) 55.15/17.12
active(length(z0)) → length(active(z0)) 55.15/17.12
inf(mark(z0)) → mark(inf(z0)) 55.15/17.12
inf(ok(z0)) → ok(inf(z0)) 55.15/17.12
take(mark(z0), z1) → mark(take(z0, z1)) 55.15/17.12
take(z0, mark(z1)) → mark(take(z0, z1)) 55.15/17.12
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 55.15/17.12
length(mark(z0)) → mark(length(z0)) 55.15/17.12
length(ok(z0)) → ok(length(z0)) 55.15/17.12
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 55.15/17.12
proper(0) → ok(0) 55.15/17.12
proper(true) → ok(true) 55.15/17.12
proper(s(z0)) → s(proper(z0)) 55.15/17.12
proper(false) → ok(false) 55.15/17.12
proper(inf(z0)) → inf(proper(z0)) 55.15/17.12
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 55.15/17.12
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 55.15/17.12
proper(nil) → ok(nil) 55.15/17.12
proper(length(z0)) → length(proper(z0)) 55.15/17.12
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 55.15/17.12
s(ok(z0)) → ok(s(z0)) 55.15/17.12
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 55.15/17.12
top(mark(z0)) → top(proper(z0)) 55.15/17.12
top(ok(z0)) → top(active(z0)) 55.15/17.12
any(z0) → s(z0) 55.15/17.12
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.12
INF(ok(z0)) → c13(INF(z0)) 55.15/17.12
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.12
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.12
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.12
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.12
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.12
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.12
S(ok(z0)) → c30(S(z0)) 55.15/17.12
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.12
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 55.15/17.12
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 55.15/17.12
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 55.15/17.12
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 55.15/17.12
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 55.15/17.12
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 55.15/17.12
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.12
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 55.15/17.12
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 55.15/17.12
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.12
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 55.15/17.12
ACTIVE(inf(z0)) → c3(S(z0)) 55.15/17.12
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 55.15/17.12
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 55.15/17.12
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 55.15/17.12
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 55.15/17.12
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 55.15/17.12
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 55.15/17.12
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 55.15/17.12
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 55.15/17.12
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 55.15/17.12
TOP(mark(0)) → c32(TOP(ok(0))) 55.15/17.12
TOP(mark(true)) → c32(TOP(ok(true))) 55.15/17.12
TOP(mark(false)) → c32(TOP(ok(false))) 55.15/17.12
TOP(mark(nil)) → c32(TOP(ok(nil)))
S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 55.15/17.12
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 55.15/17.12
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 55.15/17.12
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 55.15/17.12
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 55.15/17.12
INF(mark(z0)) → c12(INF(z0)) 55.15/17.12
INF(ok(z0)) → c13(INF(z0)) 55.15/17.12
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.12
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.12
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.12
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.12
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.12
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.12
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 55.15/17.12
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 55.15/17.12
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.12
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 55.15/17.12
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.12
S(ok(z0)) → c30(S(z0)) 55.15/17.12
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 55.15/17.12
ACTIVE(inf(z0)) → c3(S(z0)) 55.15/17.12
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 55.15/17.12
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 55.15/17.12
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 55.15/17.12
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 55.15/17.12
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 55.15/17.12
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 55.15/17.12
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 55.15/17.12
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
K tuples:

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.12
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 55.15/17.12
TOP(mark(0)) → c32(TOP(ok(0))) 55.15/17.12
TOP(mark(true)) → c32(TOP(ok(true))) 55.15/17.12
TOP(mark(false)) → c32(TOP(ok(false))) 55.15/17.12
TOP(mark(nil)) → c32(TOP(ok(nil)))
Defined Rule Symbols:

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

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, c8, c9, c10, c11, c19, c22, c24, c26, c28, c3, c, c32, c32

55.15/17.12
55.15/17.12

(21) 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))) 55.15/17.12
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 55.15/17.12
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1))) 55.15/17.12
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 55.15/17.12
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0))) 55.15/17.12
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 55.15/17.12
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil))) 55.15/17.12
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 55.15/17.12
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0))) 55.15/17.12
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1))) 55.15/17.12
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1))) 55.15/17.12
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))
55.15/17.12
55.15/17.12

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 55.15/17.12
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 55.15/17.12
active(eq(z0, z1)) → mark(false) 55.15/17.12
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 55.15/17.12
active(take(0, z0)) → mark(nil) 55.15/17.12
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 55.15/17.12
active(length(nil)) → mark(0) 55.15/17.12
active(length(cons(z0, z1))) → mark(s(length(z1))) 55.15/17.12
active(inf(z0)) → inf(active(z0)) 55.15/17.12
active(take(z0, z1)) → take(active(z0), z1) 55.15/17.12
active(take(z0, z1)) → take(z0, active(z1)) 55.15/17.12
active(length(z0)) → length(active(z0)) 55.15/17.12
inf(mark(z0)) → mark(inf(z0)) 55.15/17.12
inf(ok(z0)) → ok(inf(z0)) 55.15/17.12
take(mark(z0), z1) → mark(take(z0, z1)) 55.15/17.12
take(z0, mark(z1)) → mark(take(z0, z1)) 55.15/17.12
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 55.15/17.12
length(mark(z0)) → mark(length(z0)) 55.15/17.12
length(ok(z0)) → ok(length(z0)) 55.15/17.12
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 55.15/17.12
proper(0) → ok(0) 55.15/17.12
proper(true) → ok(true) 55.15/17.12
proper(s(z0)) → s(proper(z0)) 55.15/17.12
proper(false) → ok(false) 55.15/17.12
proper(inf(z0)) → inf(proper(z0)) 55.15/17.12
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 55.15/17.12
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 55.15/17.12
proper(nil) → ok(nil) 55.15/17.12
proper(length(z0)) → length(proper(z0)) 55.15/17.13
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 55.15/17.13
s(ok(z0)) → ok(s(z0)) 55.15/17.13
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 55.15/17.13
top(mark(z0)) → top(proper(z0)) 55.15/17.13
top(ok(z0)) → top(active(z0)) 55.15/17.13
any(z0) → s(z0) 55.15/17.13
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0)) 55.15/17.13
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.13
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 55.15/17.13
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 55.15/17.13
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 55.15/17.13
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 55.15/17.13
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 55.15/17.13
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.13
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 55.15/17.13
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 55.15/17.13
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.13
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 55.15/17.13
ACTIVE(inf(z0)) → c3(S(z0)) 55.15/17.13
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 55.15/17.13
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 55.15/17.13
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 55.15/17.13
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 55.15/17.13
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 55.15/17.13
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 55.15/17.13
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 55.15/17.13
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 55.15/17.13
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 55.15/17.13
TOP(mark(0)) → c32(TOP(ok(0))) 55.15/17.13
TOP(mark(true)) → c32(TOP(ok(true))) 55.15/17.13
TOP(mark(false)) → c32(TOP(ok(false))) 55.15/17.13
TOP(mark(nil)) → c32(TOP(ok(nil))) 55.15/17.13
TOP(ok(eq(0, 0))) → c33(TOP(mark(true)), ACTIVE(eq(0, 0))) 55.15/17.13
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 55.15/17.13
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1))) 55.15/17.13
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 55.15/17.13
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0))) 55.15/17.13
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 55.15/17.13
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil))) 55.15/17.13
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 55.15/17.13
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0))) 55.15/17.13
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1))) 55.15/17.13
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1))) 55.15/17.13
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))
S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 55.15/17.13
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 55.15/17.13
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 55.15/17.13
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 55.15/17.13
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 55.15/17.13
INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.13
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 55.15/17.13
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 55.15/17.13
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.13
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0)) 55.15/17.13
ACTIVE(inf(z0)) → c3(S(z0)) 55.15/17.13
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 55.15/17.13
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 55.15/17.13
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 55.15/17.13
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 55.15/17.13
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 55.15/17.13
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 55.15/17.13
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 55.15/17.13
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 55.15/17.13
TOP(ok(eq(0, 0))) → c33(TOP(mark(true)), ACTIVE(eq(0, 0))) 55.15/17.13
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 55.15/17.13
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1))) 55.15/17.13
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 55.15/17.13
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0))) 55.15/17.13
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 55.15/17.13
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil))) 55.15/17.13
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 55.15/17.13
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0))) 55.15/17.13
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1))) 55.15/17.13
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1))) 55.15/17.13
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))
K tuples:

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.13
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 55.15/17.13
TOP(mark(0)) → c32(TOP(ok(0))) 55.15/17.13
TOP(mark(true)) → c32(TOP(ok(true))) 55.15/17.13
TOP(mark(false)) → c32(TOP(ok(false))) 55.15/17.13
TOP(mark(nil)) → c32(TOP(ok(nil)))
Defined Rule Symbols:

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

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, c8, c9, c10, c11, c19, c22, c24, c26, c28, c3, c, c32, c32, c33

55.15/17.13
55.15/17.13

(23) 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)) 55.15/17.13
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 55.15/17.13
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 55.15/17.13
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 55.15/17.13
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 55.15/17.13
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.13
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 55.15/17.13
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 55.15/17.13
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 55.15/17.13
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 55.15/17.13
ACTIVE(inf(z0)) → c3(S(z0)) 55.15/17.13
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 55.15/17.13
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 55.15/17.13
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 55.15/17.13
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 55.15/17.13
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 55.15/17.13
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 55.15/17.13
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 55.15/17.13
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 55.15/17.13
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 55.15/17.13
TOP(ok(eq(0, 0))) → c33(TOP(mark(true)), ACTIVE(eq(0, 0))) 55.15/17.13
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 55.15/17.13
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1))) 55.15/17.13
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 55.15/17.13
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0))) 55.15/17.13
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 55.15/17.13
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil))) 55.15/17.13
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 55.15/17.13
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0))) 55.15/17.13
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1))) 55.15/17.13
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1))) 55.15/17.13
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))
55.15/17.13
55.15/17.13

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 55.15/17.13
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 55.15/17.13
active(eq(z0, z1)) → mark(false) 55.15/17.13
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 55.15/17.13
active(take(0, z0)) → mark(nil) 55.15/17.13
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 55.15/17.13
active(length(nil)) → mark(0) 55.15/17.13
active(length(cons(z0, z1))) → mark(s(length(z1))) 55.15/17.13
active(inf(z0)) → inf(active(z0)) 55.15/17.13
active(take(z0, z1)) → take(active(z0), z1) 55.15/17.13
active(take(z0, z1)) → take(z0, active(z1)) 55.15/17.13
active(length(z0)) → length(active(z0)) 55.15/17.13
inf(mark(z0)) → mark(inf(z0)) 55.15/17.13
inf(ok(z0)) → ok(inf(z0)) 55.15/17.13
take(mark(z0), z1) → mark(take(z0, z1)) 55.15/17.13
take(z0, mark(z1)) → mark(take(z0, z1)) 55.15/17.13
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 55.15/17.13
length(mark(z0)) → mark(length(z0)) 55.15/17.13
length(ok(z0)) → ok(length(z0)) 55.15/17.13
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 55.15/17.13
proper(0) → ok(0) 55.15/17.13
proper(true) → ok(true) 55.15/17.13
proper(s(z0)) → s(proper(z0)) 55.15/17.13
proper(false) → ok(false) 55.15/17.13
proper(inf(z0)) → inf(proper(z0)) 55.15/17.13
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 55.15/17.13
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 55.15/17.13
proper(nil) → ok(nil) 55.15/17.13
proper(length(z0)) → length(proper(z0)) 55.15/17.13
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 55.15/17.13
s(ok(z0)) → ok(s(z0)) 55.15/17.13
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 55.15/17.13
top(mark(z0)) → top(proper(z0)) 55.15/17.13
top(ok(z0)) → top(active(z0)) 55.15/17.13
any(z0) → s(z0) 55.15/17.13
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0)) 55.15/17.13
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.13
TOP(mark(0)) → c32(TOP(ok(0))) 55.15/17.13
TOP(mark(true)) → c32(TOP(ok(true))) 55.15/17.13
TOP(mark(false)) → c32(TOP(ok(false))) 55.15/17.13
TOP(mark(nil)) → c32(TOP(ok(nil)))
S tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0))
K tuples:

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.13
TOP(mark(0)) → c32(TOP(ok(0))) 55.15/17.13
TOP(mark(true)) → c32(TOP(ok(true))) 55.15/17.13
TOP(mark(false)) → c32(TOP(ok(false))) 55.15/17.13
TOP(mark(nil)) → c32(TOP(ok(nil)))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

55.15/17.13
55.15/17.13

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

Removed 4 trailing tuple parts
55.15/17.13
55.15/17.13

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 55.15/17.13
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 55.15/17.13
active(eq(z0, z1)) → mark(false) 55.15/17.13
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 55.15/17.13
active(take(0, z0)) → mark(nil) 55.15/17.13
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 55.15/17.13
active(length(nil)) → mark(0) 55.15/17.13
active(length(cons(z0, z1))) → mark(s(length(z1))) 55.15/17.13
active(inf(z0)) → inf(active(z0)) 55.15/17.13
active(take(z0, z1)) → take(active(z0), z1) 55.15/17.13
active(take(z0, z1)) → take(z0, active(z1)) 55.15/17.13
active(length(z0)) → length(active(z0)) 55.15/17.13
inf(mark(z0)) → mark(inf(z0)) 55.15/17.13
inf(ok(z0)) → ok(inf(z0)) 55.15/17.13
take(mark(z0), z1) → mark(take(z0, z1)) 55.15/17.13
take(z0, mark(z1)) → mark(take(z0, z1)) 55.15/17.13
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 55.15/17.13
length(mark(z0)) → mark(length(z0)) 55.15/17.13
length(ok(z0)) → ok(length(z0)) 55.15/17.13
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 55.15/17.13
proper(0) → ok(0) 55.15/17.13
proper(true) → ok(true) 55.15/17.13
proper(s(z0)) → s(proper(z0)) 55.15/17.13
proper(false) → ok(false) 55.15/17.13
proper(inf(z0)) → inf(proper(z0)) 55.15/17.13
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 55.15/17.13
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 55.15/17.13
proper(nil) → ok(nil) 55.15/17.13
proper(length(z0)) → length(proper(z0)) 55.15/17.13
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 55.15/17.13
s(ok(z0)) → ok(s(z0)) 55.15/17.13
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 55.15/17.13
top(mark(z0)) → top(proper(z0)) 55.15/17.13
top(ok(z0)) → top(active(z0)) 55.15/17.13
any(z0) → s(z0) 55.15/17.13
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0)) 55.15/17.13
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.13
TOP(mark(0)) → c32 55.15/17.13
TOP(mark(true)) → c32 55.15/17.13
TOP(mark(false)) → c32 55.15/17.13
TOP(mark(nil)) → c32
S tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0))
K tuples:

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.13
TOP(mark(0)) → c32 55.15/17.13
TOP(mark(true)) → c32 55.15/17.13
TOP(mark(false)) → c32 55.15/17.13
TOP(mark(nil)) → c32
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

55.15/17.13
55.15/17.13

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

Removed 4 trailing nodes:

TOP(mark(false)) → c32 55.15/17.13
TOP(mark(nil)) → c32 55.15/17.13
TOP(mark(true)) → c32 55.15/17.13
TOP(mark(0)) → c32
55.15/17.13
55.15/17.13

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 55.15/17.13
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 55.15/17.13
active(eq(z0, z1)) → mark(false) 55.15/17.13
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 55.15/17.13
active(take(0, z0)) → mark(nil) 55.15/17.13
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 55.15/17.13
active(length(nil)) → mark(0) 55.15/17.13
active(length(cons(z0, z1))) → mark(s(length(z1))) 55.15/17.13
active(inf(z0)) → inf(active(z0)) 55.15/17.13
active(take(z0, z1)) → take(active(z0), z1) 55.15/17.13
active(take(z0, z1)) → take(z0, active(z1)) 55.15/17.13
active(length(z0)) → length(active(z0)) 55.15/17.13
inf(mark(z0)) → mark(inf(z0)) 55.15/17.13
inf(ok(z0)) → ok(inf(z0)) 55.15/17.13
take(mark(z0), z1) → mark(take(z0, z1)) 55.15/17.13
take(z0, mark(z1)) → mark(take(z0, z1)) 55.15/17.13
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 55.15/17.13
length(mark(z0)) → mark(length(z0)) 55.15/17.13
length(ok(z0)) → ok(length(z0)) 55.15/17.13
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 55.15/17.13
proper(0) → ok(0) 55.15/17.13
proper(true) → ok(true) 55.15/17.13
proper(s(z0)) → s(proper(z0)) 55.15/17.13
proper(false) → ok(false) 55.15/17.13
proper(inf(z0)) → inf(proper(z0)) 55.15/17.13
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 55.15/17.13
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 55.15/17.13
proper(nil) → ok(nil) 55.15/17.13
proper(length(z0)) → length(proper(z0)) 55.15/17.13
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 55.15/17.13
s(ok(z0)) → ok(s(z0)) 55.15/17.13
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 55.15/17.13
top(mark(z0)) → top(proper(z0)) 55.15/17.13
top(ok(z0)) → top(active(z0)) 55.15/17.13
any(z0) → s(z0) 55.15/17.13
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0)) 55.15/17.13
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
S tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0))
K tuples:

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
Defined Rule Symbols:

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

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

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

55.15/17.13
55.15/17.13

(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(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0)) 55.15/17.13
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 55.15/17.13

POL(CONS(x1, x2)) = [3]x1 + [3]x2    55.15/17.13
POL(EQ(x1, x2)) = x1    55.15/17.13
POL(INF(x1)) = 0    55.15/17.13
POL(LENGTH(x1)) = [2]x1    55.15/17.13
POL(S(x1)) = 0    55.15/17.13
POL(TAKE(x1, x2)) = [2]x1    55.15/17.13
POL(c12(x1)) = x1    55.15/17.13
POL(c13(x1)) = x1    55.15/17.13
POL(c14(x1)) = x1    55.15/17.13
POL(c15(x1)) = x1    55.15/17.13
POL(c16(x1)) = x1    55.15/17.13
POL(c17(x1)) = x1    55.15/17.13
POL(c18(x1)) = x1    55.15/17.13
POL(c29(x1)) = x1    55.15/17.13
POL(c30(x1)) = x1    55.15/17.13
POL(c31(x1)) = x1    55.15/17.13
POL(mark(x1)) = [1] + x1    55.15/17.13
POL(ok(x1)) = [1] + x1   
55.15/17.13
55.15/17.13

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 55.15/17.13
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 55.15/17.13
active(eq(z0, z1)) → mark(false) 55.15/17.13
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 55.15/17.13
active(take(0, z0)) → mark(nil) 55.15/17.13
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 55.15/17.13
active(length(nil)) → mark(0) 55.15/17.13
active(length(cons(z0, z1))) → mark(s(length(z1))) 55.15/17.13
active(inf(z0)) → inf(active(z0)) 55.15/17.13
active(take(z0, z1)) → take(active(z0), z1) 55.15/17.13
active(take(z0, z1)) → take(z0, active(z1)) 55.15/17.13
active(length(z0)) → length(active(z0)) 55.15/17.13
inf(mark(z0)) → mark(inf(z0)) 55.15/17.13
inf(ok(z0)) → ok(inf(z0)) 55.15/17.13
take(mark(z0), z1) → mark(take(z0, z1)) 55.15/17.13
take(z0, mark(z1)) → mark(take(z0, z1)) 55.15/17.13
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 55.15/17.13
length(mark(z0)) → mark(length(z0)) 55.15/17.13
length(ok(z0)) → ok(length(z0)) 55.15/17.13
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 55.15/17.13
proper(0) → ok(0) 55.15/17.13
proper(true) → ok(true) 55.15/17.13
proper(s(z0)) → s(proper(z0)) 55.15/17.13
proper(false) → ok(false) 55.15/17.13
proper(inf(z0)) → inf(proper(z0)) 55.15/17.13
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 55.15/17.13
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 55.15/17.13
proper(nil) → ok(nil) 55.15/17.13
proper(length(z0)) → length(proper(z0)) 55.15/17.13
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 55.15/17.13
s(ok(z0)) → ok(s(z0)) 55.15/17.13
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 55.15/17.13
top(mark(z0)) → top(proper(z0)) 55.15/17.13
top(ok(z0)) → top(active(z0)) 55.15/17.13
any(z0) → s(z0) 55.15/17.13
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0)) 55.15/17.13
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
S tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0))
K tuples:

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
Defined Rule Symbols:

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

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

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

55.15/17.13
55.15/17.13

(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.

INF(mark(z0)) → c12(INF(z0))
We considered the (Usable) Rules:none
And the Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0)) 55.15/17.13
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 55.15/17.13

POL(CONS(x1, x2)) = [3]x1 + [5]x2    55.15/17.13
POL(EQ(x1, x2)) = [5]x1 + [5]x2    55.15/17.13
POL(INF(x1)) = [4]x1    55.15/17.13
POL(LENGTH(x1)) = [3]x1    55.15/17.13
POL(S(x1)) = 0    55.15/17.13
POL(TAKE(x1, x2)) = [3]x1    55.15/17.13
POL(c12(x1)) = x1    55.15/17.13
POL(c13(x1)) = x1    55.15/17.13
POL(c14(x1)) = x1    55.15/17.13
POL(c15(x1)) = x1    55.15/17.13
POL(c16(x1)) = x1    55.15/17.13
POL(c17(x1)) = x1    55.15/17.13
POL(c18(x1)) = x1    55.15/17.13
POL(c29(x1)) = x1    55.15/17.13
POL(c30(x1)) = x1    55.15/17.13
POL(c31(x1)) = x1    55.15/17.13
POL(mark(x1)) = [1] + x1    55.15/17.13
POL(ok(x1)) = x1   
55.15/17.13
55.15/17.13

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 55.15/17.13
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 55.15/17.13
active(eq(z0, z1)) → mark(false) 55.15/17.13
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 55.15/17.13
active(take(0, z0)) → mark(nil) 55.15/17.13
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 55.15/17.13
active(length(nil)) → mark(0) 55.15/17.13
active(length(cons(z0, z1))) → mark(s(length(z1))) 55.15/17.13
active(inf(z0)) → inf(active(z0)) 55.15/17.13
active(take(z0, z1)) → take(active(z0), z1) 55.15/17.13
active(take(z0, z1)) → take(z0, active(z1)) 55.15/17.13
active(length(z0)) → length(active(z0)) 55.15/17.13
inf(mark(z0)) → mark(inf(z0)) 55.15/17.13
inf(ok(z0)) → ok(inf(z0)) 55.15/17.13
take(mark(z0), z1) → mark(take(z0, z1)) 55.15/17.13
take(z0, mark(z1)) → mark(take(z0, z1)) 55.15/17.13
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 55.15/17.13
length(mark(z0)) → mark(length(z0)) 55.15/17.13
length(ok(z0)) → ok(length(z0)) 55.15/17.13
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 55.15/17.13
proper(0) → ok(0) 55.15/17.13
proper(true) → ok(true) 55.15/17.13
proper(s(z0)) → s(proper(z0)) 55.15/17.13
proper(false) → ok(false) 55.15/17.13
proper(inf(z0)) → inf(proper(z0)) 55.15/17.13
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 55.15/17.13
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 55.15/17.13
proper(nil) → ok(nil) 55.15/17.13
proper(length(z0)) → length(proper(z0)) 55.15/17.13
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 55.15/17.13
s(ok(z0)) → ok(s(z0)) 55.15/17.13
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 55.15/17.13
top(mark(z0)) → top(proper(z0)) 55.15/17.13
top(ok(z0)) → top(active(z0)) 55.15/17.13
any(z0) → s(z0) 55.15/17.13
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0)) 55.15/17.13
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
S tuples:

INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0))
K tuples:

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
INF(mark(z0)) → c12(INF(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

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

55.15/17.13
55.15/17.13

(33) 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.

INF(ok(z0)) → c13(INF(z0))
We considered the (Usable) Rules:none
And the Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0)) 55.15/17.13
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 55.15/17.13

POL(CONS(x1, x2)) = [3]x1 + [5]x2    55.15/17.13
POL(EQ(x1, x2)) = [3]x1 + [5]x2    55.15/17.13
POL(INF(x1)) = x1    55.15/17.13
POL(LENGTH(x1)) = [5]x1    55.15/17.13
POL(S(x1)) = 0    55.15/17.13
POL(TAKE(x1, x2)) = [3]x1    55.15/17.13
POL(c12(x1)) = x1    55.15/17.13
POL(c13(x1)) = x1    55.15/17.13
POL(c14(x1)) = x1    55.15/17.13
POL(c15(x1)) = x1    55.15/17.13
POL(c16(x1)) = x1    55.15/17.13
POL(c17(x1)) = x1    55.15/17.13
POL(c18(x1)) = x1    55.15/17.13
POL(c29(x1)) = x1    55.15/17.13
POL(c30(x1)) = x1    55.15/17.13
POL(c31(x1)) = x1    55.15/17.13
POL(mark(x1)) = [1] + x1    55.15/17.13
POL(ok(x1)) = [2] + x1   
55.15/17.13
55.15/17.13

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 55.15/17.13
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 55.15/17.13
active(eq(z0, z1)) → mark(false) 55.15/17.13
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 55.15/17.13
active(take(0, z0)) → mark(nil) 55.15/17.13
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 55.15/17.13
active(length(nil)) → mark(0) 55.15/17.13
active(length(cons(z0, z1))) → mark(s(length(z1))) 55.15/17.13
active(inf(z0)) → inf(active(z0)) 55.15/17.13
active(take(z0, z1)) → take(active(z0), z1) 55.15/17.13
active(take(z0, z1)) → take(z0, active(z1)) 55.15/17.13
active(length(z0)) → length(active(z0)) 55.15/17.13
inf(mark(z0)) → mark(inf(z0)) 55.15/17.13
inf(ok(z0)) → ok(inf(z0)) 55.15/17.13
take(mark(z0), z1) → mark(take(z0, z1)) 55.15/17.13
take(z0, mark(z1)) → mark(take(z0, z1)) 55.15/17.13
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 55.15/17.13
length(mark(z0)) → mark(length(z0)) 55.15/17.13
length(ok(z0)) → ok(length(z0)) 55.15/17.13
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 55.15/17.13
proper(0) → ok(0) 55.15/17.13
proper(true) → ok(true) 55.15/17.13
proper(s(z0)) → s(proper(z0)) 55.15/17.13
proper(false) → ok(false) 55.15/17.13
proper(inf(z0)) → inf(proper(z0)) 55.15/17.13
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 55.15/17.13
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 55.15/17.13
proper(nil) → ok(nil) 55.15/17.13
proper(length(z0)) → length(proper(z0)) 55.15/17.13
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 55.15/17.13
s(ok(z0)) → ok(s(z0)) 55.15/17.13
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 55.15/17.13
top(mark(z0)) → top(proper(z0)) 55.15/17.13
top(ok(z0)) → top(active(z0)) 55.15/17.13
any(z0) → s(z0) 55.15/17.13
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0)) 55.15/17.13
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
S tuples:

TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0))
K tuples:

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

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

55.15/17.13
55.15/17.13

(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.

TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0))
We considered the (Usable) Rules:none
And the Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0)) 55.15/17.13
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 55.15/17.13

POL(CONS(x1, x2)) = [3]x1 + [5]x2    55.15/17.13
POL(EQ(x1, x2)) = [5]x1 + [3]x2    55.15/17.13
POL(INF(x1)) = [3]x1    55.15/17.13
POL(LENGTH(x1)) = [3]x1    55.15/17.13
POL(S(x1)) = x1    55.15/17.13
POL(TAKE(x1, x2)) = [3]x1 + [2]x2    55.15/17.13
POL(c12(x1)) = x1    55.15/17.13
POL(c13(x1)) = x1    55.15/17.13
POL(c14(x1)) = x1    55.15/17.13
POL(c15(x1)) = x1    55.15/17.13
POL(c16(x1)) = x1    55.15/17.13
POL(c17(x1)) = x1    55.15/17.13
POL(c18(x1)) = x1    55.15/17.13
POL(c29(x1)) = x1    55.15/17.13
POL(c30(x1)) = x1    55.15/17.13
POL(c31(x1)) = x1    55.15/17.13
POL(mark(x1)) = [1] + x1    55.15/17.13
POL(ok(x1)) = [1] + x1   
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(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 55.15/17.13
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 55.15/17.13
active(eq(z0, z1)) → mark(false) 55.15/17.13
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 55.15/17.13
active(take(0, z0)) → mark(nil) 55.15/17.13
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 55.15/17.13
active(length(nil)) → mark(0) 55.15/17.13
active(length(cons(z0, z1))) → mark(s(length(z1))) 55.15/17.13
active(inf(z0)) → inf(active(z0)) 55.15/17.13
active(take(z0, z1)) → take(active(z0), z1) 55.15/17.13
active(take(z0, z1)) → take(z0, active(z1)) 55.15/17.13
active(length(z0)) → length(active(z0)) 55.15/17.13
inf(mark(z0)) → mark(inf(z0)) 55.15/17.13
inf(ok(z0)) → ok(inf(z0)) 55.15/17.13
take(mark(z0), z1) → mark(take(z0, z1)) 55.15/17.13
take(z0, mark(z1)) → mark(take(z0, z1)) 55.15/17.13
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 55.15/17.13
length(mark(z0)) → mark(length(z0)) 55.15/17.13
length(ok(z0)) → ok(length(z0)) 55.15/17.13
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 55.15/17.13
proper(0) → ok(0) 55.15/17.13
proper(true) → ok(true) 55.15/17.13
proper(s(z0)) → s(proper(z0)) 55.15/17.13
proper(false) → ok(false) 55.15/17.13
proper(inf(z0)) → inf(proper(z0)) 55.15/17.13
proper(cons(any(z0), z1)) → cons(any(any(proper(z0))), any(proper(z1))) 55.15/17.13
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 55.15/17.13
proper(nil) → ok(nil) 55.15/17.13
proper(length(z0)) → length(proper(z0)) 55.15/17.13
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 55.15/17.13
s(ok(z0)) → ok(s(z0)) 55.15/17.13
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 55.15/17.13
top(mark(z0)) → top(proper(z0)) 55.15/17.13
top(ok(z0)) → top(active(z0)) 55.15/17.13
any(z0) → s(z0) 55.15/17.13
any(proper(z0)) → any(any(any(z0)))
Tuples:

INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0)) 55.15/17.13
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
S tuples:none
K tuples:

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 55.15/17.13
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 55.15/17.13
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 55.15/17.13
LENGTH(mark(z0)) → c17(LENGTH(z0)) 55.15/17.13
LENGTH(ok(z0)) → c18(LENGTH(z0)) 55.15/17.13
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 55.15/17.13
INF(mark(z0)) → c12(INF(z0)) 55.15/17.13
INF(ok(z0)) → c13(INF(z0)) 55.15/17.13
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 55.15/17.13
S(ok(z0)) → c30(S(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

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

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

The set S is empty
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(38) BOUNDS(O(1), O(1))

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55.39/17.26 EOF