218.38/71.61 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml

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

0 CpxTRS

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

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↳2 CdtProblem

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

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↳4 CdtProblem

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↳5 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

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↳6 CdtProblem

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

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↳8 CdtProblem

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

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↳10 CdtProblem

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↳11 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

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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 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

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↳18 CdtProblem

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

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↳20 CdtProblem

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↳21 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳22 CdtProblem

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

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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 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳28 CdtProblem

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

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↳30 CdtProblem

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

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↳32 CdtProblem

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↳33 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳34 CdtProblem

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↳35 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳36 CdtProblem

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

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↳38 CdtProblem

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

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↳40 CdtProblem

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↳41 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳42 CdtProblem

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↳43 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳44 CdtProblem

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↳45 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳46 CdtProblem

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↳47 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳48 CdtProblem

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↳49 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳50 CdtProblem

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

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↳52 CdtProblem

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↳53 CdtUnreachableProof (⇔)

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↳54 CdtProblem

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↳55 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳56 CdtProblem

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↳57 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳58 CdtProblem

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↳59 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳60 CdtProblem

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↳61 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳62 CdtProblem

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↳63 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳64 CdtProblem

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↳65 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

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↳66 CdtProblem

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↳67 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳68 CdtProblem

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↳69 SIsEmptyProof (BOTH BOUNDS(ID, ID))

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↳70 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) 218.77/71.76
active(eq(s(X), s(Y))) → mark(eq(X, Y)) 218.77/71.76
active(eq(X, Y)) → mark(false) 218.77/71.76
active(inf(X)) → mark(cons(X, inf(s(X)))) 218.77/71.76
active(take(0, X)) → mark(nil) 218.77/71.76
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L))) 218.77/71.76
active(length(nil)) → mark(0) 218.77/71.76
active(length(cons(X, L))) → mark(s(length(L))) 218.77/71.76
active(inf(X)) → inf(active(X)) 218.77/71.76
active(take(X1, X2)) → take(active(X1), X2) 218.77/71.76
active(take(X1, X2)) → take(X1, active(X2)) 218.77/71.76
active(length(X)) → length(active(X)) 218.77/71.76
inf(mark(X)) → mark(inf(X)) 218.77/71.76
take(mark(X1), X2) → mark(take(X1, X2)) 218.77/71.76
take(X1, mark(X2)) → mark(take(X1, X2)) 218.77/71.76
length(mark(X)) → mark(length(X)) 218.77/71.76
proper(eq(X1, X2)) → eq(proper(X1), proper(X2)) 218.77/71.76
proper(0) → ok(0) 218.77/71.76
proper(true) → ok(true) 218.77/71.76
proper(s(X)) → s(proper(X)) 218.77/71.76
proper(false) → ok(false) 218.77/71.76
proper(inf(X)) → inf(proper(X)) 218.77/71.76
proper(cons(X1, X2)) → cons(proper(X1), proper(X2)) 218.77/71.76
proper(take(X1, X2)) → take(proper(X1), proper(X2)) 218.77/71.77
proper(nil) → ok(nil) 218.77/71.77
proper(length(X)) → length(proper(X)) 218.77/71.77
eq(ok(X1), ok(X2)) → ok(eq(X1, X2)) 218.77/71.77
s(ok(X)) → ok(s(X)) 218.77/71.77
inf(ok(X)) → ok(inf(X)) 218.77/71.77
cons(ok(X1), ok(X2)) → ok(cons(X1, X2)) 218.77/71.77
take(ok(X1), ok(X2)) → ok(take(X1, X2)) 218.77/71.77
length(ok(X)) → ok(length(X)) 218.77/71.77
top(mark(X)) → top(proper(X)) 218.77/71.77
top(ok(X)) → top(active(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) 218.77/71.77
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 218.77/71.77
active(eq(z0, z1)) → mark(false) 218.77/71.77
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 218.77/71.77
active(take(0, z0)) → mark(nil) 218.77/71.77
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 218.77/71.77
active(length(nil)) → mark(0) 218.77/71.77
active(length(cons(z0, z1))) → mark(s(length(z1))) 218.77/71.77
active(inf(z0)) → inf(active(z0)) 218.77/71.77
active(take(z0, z1)) → take(active(z0), z1) 218.77/71.77
active(take(z0, z1)) → take(z0, active(z1)) 218.77/71.77
active(length(z0)) → length(active(z0)) 218.77/71.77
inf(mark(z0)) → mark(inf(z0)) 218.77/71.77
inf(ok(z0)) → ok(inf(z0)) 218.77/71.77
take(mark(z0), z1) → mark(take(z0, z1)) 218.77/71.77
take(z0, mark(z1)) → mark(take(z0, z1)) 218.77/71.77
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 218.77/71.77
length(mark(z0)) → mark(length(z0)) 218.77/71.77
length(ok(z0)) → ok(length(z0)) 218.77/71.77
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 218.77/71.77
proper(0) → ok(0) 218.77/71.77
proper(true) → ok(true) 218.77/71.77
proper(s(z0)) → s(proper(z0)) 218.77/71.77
proper(false) → ok(false) 218.77/71.77
proper(inf(z0)) → inf(proper(z0)) 218.77/71.77
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 218.77/71.77
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 218.77/71.77
proper(nil) → ok(nil) 218.77/71.77
proper(length(z0)) → length(proper(z0)) 218.77/71.77
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 218.77/71.77
s(ok(z0)) → ok(s(z0)) 218.77/71.77
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 218.77/71.77
top(mark(z0)) → top(proper(z0)) 218.77/71.77
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 218.77/71.77
ACTIVE(inf(z0)) → c3(CONS(z0, inf(s(z0))), INF(s(z0)), S(z0)) 218.77/71.77
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2)) 218.77/71.77
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1)) 218.77/71.77
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 218.77/71.77
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 218.77/71.77
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 218.77/71.77
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 218.77/71.77
INF(mark(z0)) → c12(INF(z0)) 218.77/71.77
INF(ok(z0)) → c13(INF(z0)) 218.77/71.77
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 218.77/71.77
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 218.77/71.77
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 218.77/71.77
LENGTH(mark(z0)) → c17(LENGTH(z0)) 218.77/71.77
LENGTH(ok(z0)) → c18(LENGTH(z0)) 218.77/71.77
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.77
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 218.77/71.77
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 218.77/71.77
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.77
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.77
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 218.77/71.77
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 218.77/71.77
S(ok(z0)) → c30(S(z0)) 218.77/71.77
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 218.77/71.77
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 218.77/71.77
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 218.77/71.77
ACTIVE(inf(z0)) → c3(CONS(z0, inf(s(z0))), INF(s(z0)), S(z0)) 218.77/71.77
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2)) 218.77/71.77
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1)) 218.77/71.77
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 218.77/71.77
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 218.77/71.77
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 218.77/71.77
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 218.77/71.77
INF(mark(z0)) → c12(INF(z0)) 218.77/71.77
INF(ok(z0)) → c13(INF(z0)) 218.77/71.77
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 218.77/71.77
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 218.77/71.77
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 218.77/71.77
LENGTH(mark(z0)) → c17(LENGTH(z0)) 218.77/71.77
LENGTH(ok(z0)) → c18(LENGTH(z0)) 218.77/71.77
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.77
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 218.77/71.77
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 218.77/71.77
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.77
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.77
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 218.77/71.77
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 218.77/71.77
S(ok(z0)) → c30(S(z0)) 218.77/71.77
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 218.77/71.77
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 218.77/71.77
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

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

Removed 2 trailing tuple parts

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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 218.77/71.77
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 218.77/71.77
active(eq(z0, z1)) → mark(false) 218.77/71.77
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 218.77/71.77
active(take(0, z0)) → mark(nil) 218.77/71.77
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 218.77/71.77
active(length(nil)) → mark(0) 218.77/71.77
active(length(cons(z0, z1))) → mark(s(length(z1))) 218.77/71.77
active(inf(z0)) → inf(active(z0)) 218.77/71.77
active(take(z0, z1)) → take(active(z0), z1) 218.77/71.77
active(take(z0, z1)) → take(z0, active(z1)) 218.77/71.77
active(length(z0)) → length(active(z0)) 218.77/71.77
inf(mark(z0)) → mark(inf(z0)) 218.77/71.77
inf(ok(z0)) → ok(inf(z0)) 218.77/71.77
take(mark(z0), z1) → mark(take(z0, z1)) 218.77/71.77
take(z0, mark(z1)) → mark(take(z0, z1)) 218.77/71.77
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 218.77/71.77
length(mark(z0)) → mark(length(z0)) 218.77/71.77
length(ok(z0)) → ok(length(z0)) 218.77/71.77
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 218.77/71.77
proper(0) → ok(0) 218.77/71.77
proper(true) → ok(true) 218.77/71.77
proper(s(z0)) → s(proper(z0)) 218.77/71.77
proper(false) → ok(false) 218.77/71.77
proper(inf(z0)) → inf(proper(z0)) 218.77/71.77
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 218.77/71.77
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 218.77/71.77
proper(nil) → ok(nil) 218.77/71.77
proper(length(z0)) → length(proper(z0)) 218.77/71.77
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 218.77/71.77
s(ok(z0)) → ok(s(z0)) 218.77/71.77
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 218.77/71.77
top(mark(z0)) → top(proper(z0)) 218.77/71.77
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 218.77/71.77
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2)) 218.77/71.77
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1)) 218.77/71.77
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 218.77/71.77
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 218.77/71.77
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 218.77/71.77
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 218.77/71.77
INF(mark(z0)) → c12(INF(z0)) 218.77/71.77
INF(ok(z0)) → c13(INF(z0)) 218.77/71.77
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 218.77/71.77
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 218.77/71.77
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 218.77/71.77
LENGTH(mark(z0)) → c17(LENGTH(z0)) 218.77/71.77
LENGTH(ok(z0)) → c18(LENGTH(z0)) 218.77/71.77
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.77
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 218.77/71.77
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 218.77/71.77
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.77
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.77
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 218.77/71.77
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 218.77/71.77
S(ok(z0)) → c30(S(z0)) 218.77/71.77
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 218.77/71.77
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 218.77/71.77
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 218.77/71.77
ACTIVE(inf(z0)) → c3(S(z0))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 218.77/71.77
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2)) 218.77/71.77
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1)) 218.77/71.77
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 218.77/71.77
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 218.77/71.77
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 218.77/71.77
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 218.77/71.77
INF(mark(z0)) → c12(INF(z0)) 218.77/71.77
INF(ok(z0)) → c13(INF(z0)) 218.77/71.77
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 218.77/71.77
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 218.77/71.77
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 218.77/71.77
LENGTH(mark(z0)) → c17(LENGTH(z0)) 218.77/71.77
LENGTH(ok(z0)) → c18(LENGTH(z0)) 218.77/71.77
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.77
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 218.77/71.77
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 218.77/71.77
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.77
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.77
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 218.77/71.77
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 218.77/71.77
S(ok(z0)) → c30(S(z0)) 218.77/71.77
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 218.77/71.77
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 218.77/71.77
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 218.77/71.77
ACTIVE(inf(z0)) → c3(S(z0))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

218.77/71.77

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

Split RHS of tuples not part of any SCC

218.77/71.77

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 218.77/71.77
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 218.77/71.77
active(eq(z0, z1)) → mark(false) 218.77/71.77
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 218.77/71.77
active(take(0, z0)) → mark(nil) 218.77/71.77
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 218.77/71.77
active(length(nil)) → mark(0) 218.77/71.77
active(length(cons(z0, z1))) → mark(s(length(z1))) 218.77/71.77
active(inf(z0)) → inf(active(z0)) 218.77/71.77
active(take(z0, z1)) → take(active(z0), z1) 218.77/71.77
active(take(z0, z1)) → take(z0, active(z1)) 218.77/71.77
active(length(z0)) → length(active(z0)) 218.77/71.77
inf(mark(z0)) → mark(inf(z0)) 218.77/71.77
inf(ok(z0)) → ok(inf(z0)) 218.77/71.77
take(mark(z0), z1) → mark(take(z0, z1)) 218.77/71.77
take(z0, mark(z1)) → mark(take(z0, z1)) 218.77/71.77
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 218.77/71.77
length(mark(z0)) → mark(length(z0)) 218.77/71.78
length(ok(z0)) → ok(length(z0)) 218.77/71.78
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 218.77/71.78
proper(0) → ok(0) 218.77/71.78
proper(true) → ok(true) 218.77/71.78
proper(s(z0)) → s(proper(z0)) 218.77/71.78
proper(false) → ok(false) 218.77/71.78
proper(inf(z0)) → inf(proper(z0)) 218.77/71.78
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 218.77/71.78
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 218.77/71.78
proper(nil) → ok(nil) 218.77/71.78
proper(length(z0)) → length(proper(z0)) 218.77/71.78
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 218.77/71.78
s(ok(z0)) → ok(s(z0)) 218.77/71.78
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 218.77/71.78
top(mark(z0)) → top(proper(z0)) 218.77/71.78
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 218.77/71.78
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 218.77/71.78
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 218.77/71.78
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 218.77/71.78
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 218.77/71.78
INF(mark(z0)) → c12(INF(z0)) 218.77/71.78
INF(ok(z0)) → c13(INF(z0)) 218.77/71.78
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 218.77/71.78
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 218.77/71.78
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 218.77/71.78
LENGTH(mark(z0)) → c17(LENGTH(z0)) 218.77/71.78
LENGTH(ok(z0)) → c18(LENGTH(z0)) 218.77/71.78
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.78
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 218.77/71.78
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 218.77/71.78
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.78
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.78
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 218.77/71.78
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 218.77/71.78
S(ok(z0)) → c30(S(z0)) 218.77/71.78
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 218.77/71.78
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 218.77/71.78
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 218.77/71.78
ACTIVE(inf(z0)) → c3(S(z0)) 218.77/71.78
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 218.77/71.78
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 218.77/71.78
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 218.77/71.78
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 218.77/71.78
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) 218.77/71.78
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 218.77/71.78
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 218.77/71.78
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 218.77/71.78
INF(mark(z0)) → c12(INF(z0)) 218.77/71.78
INF(ok(z0)) → c13(INF(z0)) 218.77/71.78
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 218.77/71.78
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 218.77/71.78
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 218.77/71.78
LENGTH(mark(z0)) → c17(LENGTH(z0)) 218.77/71.78
LENGTH(ok(z0)) → c18(LENGTH(z0)) 218.77/71.78
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.78
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 218.77/71.78
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 218.77/71.78
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.78
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.78
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 218.77/71.78
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 218.77/71.78
S(ok(z0)) → c30(S(z0)) 218.77/71.78
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 218.77/71.78
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 218.77/71.78
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 218.77/71.78
ACTIVE(inf(z0)) → c3(S(z0)) 218.77/71.78
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 218.77/71.78
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 218.77/71.78
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 218.77/71.78
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

218.77/71.78

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

Use narrowing to replace ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) by

ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)), ACTIVE(eq(0, 0))) 218.77/71.78
ACTIVE(inf(eq(s(z0), s(z1)))) → c8(INF(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 218.77/71.78
ACTIVE(inf(eq(z0, z1))) → c8(INF(mark(false)), ACTIVE(eq(z0, z1))) 218.77/71.78
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 218.77/71.78
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 218.77/71.78
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 218.77/71.78
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil))) 218.77/71.78
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 218.77/71.78
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 218.77/71.78
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 218.77/71.78
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 218.77/71.78
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))

218.77/71.78

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 218.77/71.78
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 218.77/71.78
active(eq(z0, z1)) → mark(false) 218.77/71.78
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 218.77/71.78
active(take(0, z0)) → mark(nil) 218.77/71.78
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 218.77/71.78
active(length(nil)) → mark(0) 218.77/71.78
active(length(cons(z0, z1))) → mark(s(length(z1))) 218.77/71.78
active(inf(z0)) → inf(active(z0)) 218.77/71.78
active(take(z0, z1)) → take(active(z0), z1) 218.77/71.78
active(take(z0, z1)) → take(z0, active(z1)) 218.77/71.78
active(length(z0)) → length(active(z0)) 218.77/71.78
inf(mark(z0)) → mark(inf(z0)) 218.77/71.78
inf(ok(z0)) → ok(inf(z0)) 218.77/71.78
take(mark(z0), z1) → mark(take(z0, z1)) 218.77/71.78
take(z0, mark(z1)) → mark(take(z0, z1)) 218.77/71.78
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 218.77/71.78
length(mark(z0)) → mark(length(z0)) 218.77/71.78
length(ok(z0)) → ok(length(z0)) 218.77/71.78
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 218.77/71.78
proper(0) → ok(0) 218.77/71.78
proper(true) → ok(true) 218.77/71.78
proper(s(z0)) → s(proper(z0)) 218.77/71.78
proper(false) → ok(false) 218.77/71.78
proper(inf(z0)) → inf(proper(z0)) 218.77/71.78
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 218.77/71.78
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 218.77/71.78
proper(nil) → ok(nil) 218.77/71.78
proper(length(z0)) → length(proper(z0)) 218.77/71.78
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 218.77/71.78
s(ok(z0)) → ok(s(z0)) 218.77/71.78
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 218.77/71.78
top(mark(z0)) → top(proper(z0)) 218.77/71.78
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 218.77/71.78
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 218.77/71.78
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 218.77/71.78
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 218.77/71.78
INF(mark(z0)) → c12(INF(z0)) 218.77/71.78
INF(ok(z0)) → c13(INF(z0)) 218.77/71.78
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 218.77/71.78
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 218.77/71.78
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 218.77/71.78
LENGTH(mark(z0)) → c17(LENGTH(z0)) 218.77/71.78
LENGTH(ok(z0)) → c18(LENGTH(z0)) 218.77/71.78
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.78
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 218.77/71.78
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 218.77/71.78
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.78
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.78
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 218.77/71.78
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 218.77/71.78
S(ok(z0)) → c30(S(z0)) 218.77/71.78
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 218.77/71.78
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 218.77/71.78
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 218.77/71.78
ACTIVE(inf(z0)) → c3(S(z0)) 218.77/71.78
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 218.77/71.78
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 218.77/71.78
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 218.77/71.78
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 218.77/71.78
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)), ACTIVE(eq(0, 0))) 218.77/71.78
ACTIVE(inf(eq(s(z0), s(z1)))) → c8(INF(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 218.77/71.78
ACTIVE(inf(eq(z0, z1))) → c8(INF(mark(false)), ACTIVE(eq(z0, z1))) 218.77/71.78
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 218.77/71.78
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 218.77/71.78
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 218.77/71.78
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil))) 218.77/71.78
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 218.77/71.78
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 218.77/71.78
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 218.77/71.78
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 218.77/71.78
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 218.77/71.78
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 218.77/71.78
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 218.77/71.78
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 218.77/71.78
INF(mark(z0)) → c12(INF(z0)) 218.77/71.78
INF(ok(z0)) → c13(INF(z0)) 218.77/71.78
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 218.77/71.78
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 218.77/71.78
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 218.77/71.78
LENGTH(mark(z0)) → c17(LENGTH(z0)) 218.77/71.78
LENGTH(ok(z0)) → c18(LENGTH(z0)) 218.77/71.78
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.78
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 218.77/71.78
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 218.77/71.78
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.78
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 218.77/71.78
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 218.77/71.78
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 218.77/71.78
S(ok(z0)) → c30(S(z0)) 218.77/71.78
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 218.77/71.78
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 218.77/71.78
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 218.77/71.78
ACTIVE(inf(z0)) → c3(S(z0)) 218.77/71.78
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.17/71.84
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.17/71.84
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.17/71.84
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.17/71.84
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)), ACTIVE(eq(0, 0))) 219.17/71.84
ACTIVE(inf(eq(s(z0), s(z1)))) → c8(INF(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 219.17/71.84
ACTIVE(inf(eq(z0, z1))) → c8(INF(mark(false)), ACTIVE(eq(z0, z1))) 219.17/71.84
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.17/71.84
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.17/71.84
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.17/71.84
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil))) 219.17/71.84
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.17/71.84
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.17/71.84
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.17/71.84
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.17/71.84
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

219.17/71.84

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

Removed 1 trailing tuple parts

219.17/71.84

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.17/71.84
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.17/71.84
active(eq(z0, z1)) → mark(false) 219.17/71.84
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.17/71.84
active(take(0, z0)) → mark(nil) 219.17/71.84
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.17/71.84
active(length(nil)) → mark(0) 219.17/71.84
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.17/71.84
active(inf(z0)) → inf(active(z0)) 219.17/71.84
active(take(z0, z1)) → take(active(z0), z1) 219.17/71.84
active(take(z0, z1)) → take(z0, active(z1)) 219.17/71.84
active(length(z0)) → length(active(z0)) 219.17/71.84
inf(mark(z0)) → mark(inf(z0)) 219.17/71.84
inf(ok(z0)) → ok(inf(z0)) 219.17/71.84
take(mark(z0), z1) → mark(take(z0, z1)) 219.17/71.84
take(z0, mark(z1)) → mark(take(z0, z1)) 219.17/71.84
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.17/71.84
length(mark(z0)) → mark(length(z0)) 219.17/71.84
length(ok(z0)) → ok(length(z0)) 219.17/71.84
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.17/71.84
proper(0) → ok(0) 219.17/71.84
proper(true) → ok(true) 219.17/71.84
proper(s(z0)) → s(proper(z0)) 219.17/71.84
proper(false) → ok(false) 219.17/71.84
proper(inf(z0)) → inf(proper(z0)) 219.17/71.84
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.17/71.84
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.17/71.84
proper(nil) → ok(nil) 219.17/71.84
proper(length(z0)) → length(proper(z0)) 219.17/71.84
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.17/71.84
s(ok(z0)) → ok(s(z0)) 219.17/71.84
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.17/71.84
top(mark(z0)) → top(proper(z0)) 219.17/71.84
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.17/71.84
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.17/71.84
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.17/71.84
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 219.17/71.84
INF(mark(z0)) → c12(INF(z0)) 219.17/71.84
INF(ok(z0)) → c13(INF(z0)) 219.17/71.84
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.17/71.84
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.17/71.84
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.17/71.84
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.17/71.84
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.17/71.84
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.84
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 219.17/71.84
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.17/71.84
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.84
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.84
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.17/71.84
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.17/71.84
S(ok(z0)) → c30(S(z0)) 219.17/71.84
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.17/71.84
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.17/71.84
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.17/71.84
ACTIVE(inf(z0)) → c3(S(z0)) 219.17/71.84
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.17/71.84
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.17/71.84
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.17/71.84
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.17/71.84
ACTIVE(inf(eq(s(z0), s(z1)))) → c8(INF(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 219.17/71.84
ACTIVE(inf(eq(z0, z1))) → c8(INF(mark(false)), ACTIVE(eq(z0, z1))) 219.17/71.84
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.17/71.84
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.17/71.84
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.17/71.84
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil))) 219.17/71.84
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.17/71.84
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.17/71.84
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.17/71.84
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.17/71.84
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.17/71.84
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.17/71.84
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.17/71.84
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.17/71.84
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 219.17/71.84
INF(mark(z0)) → c12(INF(z0)) 219.17/71.84
INF(ok(z0)) → c13(INF(z0)) 219.17/71.84
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.17/71.84
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.17/71.84
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.17/71.84
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.17/71.84
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.17/71.84
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.84
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 219.17/71.84
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.17/71.84
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.84
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.84
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.17/71.84
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.17/71.84
S(ok(z0)) → c30(S(z0)) 219.17/71.84
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.17/71.84
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.17/71.84
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.17/71.84
ACTIVE(inf(z0)) → c3(S(z0)) 219.17/71.84
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.17/71.84
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.17/71.84
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.17/71.84
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.17/71.84
ACTIVE(inf(eq(s(z0), s(z1)))) → c8(INF(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 219.17/71.84
ACTIVE(inf(eq(z0, z1))) → c8(INF(mark(false)), ACTIVE(eq(z0, z1))) 219.17/71.84
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.17/71.84
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.17/71.84
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.17/71.84
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil))) 219.17/71.84
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.17/71.84
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.17/71.84
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.17/71.84
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.17/71.84
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.17/71.84
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

219.17/71.84

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

Split RHS of tuples not part of any SCC

219.17/71.84

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.17/71.84
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.17/71.84
active(eq(z0, z1)) → mark(false) 219.17/71.84
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.17/71.84
active(take(0, z0)) → mark(nil) 219.17/71.84
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.17/71.84
active(length(nil)) → mark(0) 219.17/71.84
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.17/71.84
active(inf(z0)) → inf(active(z0)) 219.17/71.84
active(take(z0, z1)) → take(active(z0), z1) 219.17/71.84
active(take(z0, z1)) → take(z0, active(z1)) 219.17/71.84
active(length(z0)) → length(active(z0)) 219.17/71.84
inf(mark(z0)) → mark(inf(z0)) 219.17/71.84
inf(ok(z0)) → ok(inf(z0)) 219.17/71.84
take(mark(z0), z1) → mark(take(z0, z1)) 219.17/71.84
take(z0, mark(z1)) → mark(take(z0, z1)) 219.17/71.84
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.17/71.84
length(mark(z0)) → mark(length(z0)) 219.17/71.84
length(ok(z0)) → ok(length(z0)) 219.17/71.84
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.17/71.84
proper(0) → ok(0) 219.17/71.84
proper(true) → ok(true) 219.17/71.84
proper(s(z0)) → s(proper(z0)) 219.17/71.84
proper(false) → ok(false) 219.17/71.84
proper(inf(z0)) → inf(proper(z0)) 219.17/71.84
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.17/71.84
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.17/71.84
proper(nil) → ok(nil) 219.17/71.84
proper(length(z0)) → length(proper(z0)) 219.17/71.84
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.17/71.84
s(ok(z0)) → ok(s(z0)) 219.17/71.84
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.17/71.84
top(mark(z0)) → top(proper(z0)) 219.17/71.84
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.17/71.84
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.17/71.84
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.17/71.84
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 219.17/71.84
INF(mark(z0)) → c12(INF(z0)) 219.17/71.84
INF(ok(z0)) → c13(INF(z0)) 219.17/71.84
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.17/71.84
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.17/71.84
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.17/71.84
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.17/71.84
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.17/71.84
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.84
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 219.17/71.84
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.17/71.84
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.84
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.84
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.17/71.84
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.17/71.84
S(ok(z0)) → c30(S(z0)) 219.17/71.84
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.17/71.84
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.17/71.84
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.17/71.84
ACTIVE(inf(z0)) → c3(S(z0)) 219.17/71.84
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.17/71.84
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.17/71.84
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.17/71.84
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.17/71.84
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.17/71.84
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.17/71.84
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.17/71.84
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil))) 219.17/71.84
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.17/71.84
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.17/71.84
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.17/71.84
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.17/71.84
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.17/71.84
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.17/71.84
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.17/71.84
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.17/71.84
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.17/71.84
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.17/71.84
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.17/71.84
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.17/71.84
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) 219.17/71.84
INF(mark(z0)) → c12(INF(z0)) 219.17/71.84
INF(ok(z0)) → c13(INF(z0)) 219.17/71.84
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.17/71.84
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.17/71.84
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.17/71.84
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.17/71.84
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.17/71.84
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.84
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 219.17/71.84
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.17/71.84
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.84
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.84
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.17/71.84
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.17/71.84
S(ok(z0)) → c30(S(z0)) 219.17/71.84
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.17/71.84
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.17/71.84
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.17/71.84
ACTIVE(inf(z0)) → c3(S(z0)) 219.17/71.84
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.17/71.84
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.17/71.84
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.17/71.84
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.17/71.84
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.17/71.84
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.17/71.84
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.17/71.84
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil))) 219.17/71.84
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.17/71.84
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.17/71.84
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.17/71.84
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.17/71.84
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.17/71.84
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.17/71.84
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.17/71.84
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.17/71.84
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.17/71.84
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2

219.17/71.84

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

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

ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)), ACTIVE(eq(0, 0))) 219.17/71.84
ACTIVE(length(eq(s(z0), s(z1)))) → c11(LENGTH(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 219.17/71.84
ACTIVE(length(eq(z0, z1))) → c11(LENGTH(mark(false)), ACTIVE(eq(z0, z1))) 219.17/71.84
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.17/71.84
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.17/71.84
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.17/71.84
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)), ACTIVE(length(nil))) 219.17/71.84
ACTIVE(length(length(cons(z0, z1)))) → c11(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.17/71.84
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.17/71.84
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.17/71.84
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.17/71.84
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))

219.17/71.84

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.17/71.84
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.17/71.84
active(eq(z0, z1)) → mark(false) 219.17/71.84
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.17/71.84
active(take(0, z0)) → mark(nil) 219.17/71.84
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.17/71.84
active(length(nil)) → mark(0) 219.17/71.84
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.17/71.84
active(inf(z0)) → inf(active(z0)) 219.17/71.84
active(take(z0, z1)) → take(active(z0), z1) 219.17/71.84
active(take(z0, z1)) → take(z0, active(z1)) 219.17/71.85
active(length(z0)) → length(active(z0)) 219.17/71.85
inf(mark(z0)) → mark(inf(z0)) 219.17/71.85
inf(ok(z0)) → ok(inf(z0)) 219.17/71.85
take(mark(z0), z1) → mark(take(z0, z1)) 219.17/71.85
take(z0, mark(z1)) → mark(take(z0, z1)) 219.17/71.85
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.17/71.85
length(mark(z0)) → mark(length(z0)) 219.17/71.85
length(ok(z0)) → ok(length(z0)) 219.17/71.85
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.17/71.85
proper(0) → ok(0) 219.17/71.85
proper(true) → ok(true) 219.17/71.85
proper(s(z0)) → s(proper(z0)) 219.17/71.85
proper(false) → ok(false) 219.17/71.85
proper(inf(z0)) → inf(proper(z0)) 219.17/71.85
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.17/71.85
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.17/71.85
proper(nil) → ok(nil) 219.17/71.85
proper(length(z0)) → length(proper(z0)) 219.17/71.85
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.17/71.85
s(ok(z0)) → ok(s(z0)) 219.17/71.85
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.17/71.85
top(mark(z0)) → top(proper(z0)) 219.17/71.85
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.17/71.85
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.17/71.85
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.17/71.85
INF(mark(z0)) → c12(INF(z0)) 219.17/71.85
INF(ok(z0)) → c13(INF(z0)) 219.17/71.85
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.17/71.85
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.17/71.85
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.17/71.85
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.17/71.85
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.17/71.85
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.85
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 219.17/71.85
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.17/71.85
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.85
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.85
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.17/71.85
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.17/71.85
S(ok(z0)) → c30(S(z0)) 219.17/71.85
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.17/71.85
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.17/71.85
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.17/71.85
ACTIVE(inf(z0)) → c3(S(z0)) 219.17/71.85
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.17/71.85
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.17/71.85
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.17/71.85
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.17/71.85
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.17/71.85
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.17/71.85
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.17/71.85
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil))) 219.17/71.85
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.17/71.85
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.17/71.85
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.17/71.85
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.17/71.85
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.17/71.85
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.17/71.85
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.17/71.85
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.17/71.85
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.17/71.85
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.17/71.85
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)), ACTIVE(eq(0, 0))) 219.17/71.85
ACTIVE(length(eq(s(z0), s(z1)))) → c11(LENGTH(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 219.17/71.85
ACTIVE(length(eq(z0, z1))) → c11(LENGTH(mark(false)), ACTIVE(eq(z0, z1))) 219.17/71.85
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.17/71.85
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.17/71.85
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.17/71.85
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)), ACTIVE(length(nil))) 219.17/71.85
ACTIVE(length(length(cons(z0, z1)))) → c11(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.17/71.85
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.17/71.85
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.17/71.85
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.17/71.85
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.17/71.85
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.17/71.85
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.17/71.85
INF(mark(z0)) → c12(INF(z0)) 219.17/71.85
INF(ok(z0)) → c13(INF(z0)) 219.17/71.85
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.17/71.85
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.17/71.85
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.17/71.85
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.17/71.85
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.17/71.85
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.85
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 219.17/71.85
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.17/71.85
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.85
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.17/71.85
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.17/71.85
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.17/71.85
S(ok(z0)) → c30(S(z0)) 219.17/71.85
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.17/71.85
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.17/71.85
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.17/71.85
ACTIVE(inf(z0)) → c3(S(z0)) 219.17/71.85
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.17/71.85
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.17/71.85
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.17/71.85
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.17/71.85
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.17/71.85
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.17/71.85
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.17/71.85
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil))) 219.17/71.85
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.17/71.85
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.17/71.85
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)), ACTIVE(eq(0, 0))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c11(LENGTH(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c11(LENGTH(mark(false)), ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)), ACTIVE(length(nil))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c11(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11

219.56/71.99

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

Removed 3 trailing tuple parts

219.56/71.99

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.56/71.99
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.56/71.99
active(eq(z0, z1)) → mark(false) 219.56/71.99
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.56/71.99
active(take(0, z0)) → mark(nil) 219.56/71.99
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.56/71.99
active(length(nil)) → mark(0) 219.56/71.99
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.56/71.99
active(inf(z0)) → inf(active(z0)) 219.56/71.99
active(take(z0, z1)) → take(active(z0), z1) 219.56/71.99
active(take(z0, z1)) → take(z0, active(z1)) 219.56/71.99
active(length(z0)) → length(active(z0)) 219.56/71.99
inf(mark(z0)) → mark(inf(z0)) 219.56/71.99
inf(ok(z0)) → ok(inf(z0)) 219.56/71.99
take(mark(z0), z1) → mark(take(z0, z1)) 219.56/71.99
take(z0, mark(z1)) → mark(take(z0, z1)) 219.56/71.99
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.56/71.99
length(mark(z0)) → mark(length(z0)) 219.56/71.99
length(ok(z0)) → ok(length(z0)) 219.56/71.99
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.56/71.99
proper(0) → ok(0) 219.56/71.99
proper(true) → ok(true) 219.56/71.99
proper(s(z0)) → s(proper(z0)) 219.56/71.99
proper(false) → ok(false) 219.56/71.99
proper(inf(z0)) → inf(proper(z0)) 219.56/71.99
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.56/71.99
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.56/71.99
proper(nil) → ok(nil) 219.56/71.99
proper(length(z0)) → length(proper(z0)) 219.56/71.99
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.56/71.99
s(ok(z0)) → ok(s(z0)) 219.56/71.99
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.56/71.99
top(mark(z0)) → top(proper(z0)) 219.56/71.99
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/71.99
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/71.99
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/71.99
INF(mark(z0)) → c12(INF(z0)) 219.56/71.99
INF(ok(z0)) → c13(INF(z0)) 219.56/71.99
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/71.99
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/71.99
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/71.99
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/71.99
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/71.99
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/71.99
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/71.99
S(ok(z0)) → c30(S(z0)) 219.56/71.99
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/71.99
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/71.99
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/71.99
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c11(LENGTH(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c11(LENGTH(mark(false)), ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c11(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/71.99
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/71.99
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/71.99
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/71.99
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/71.99
INF(mark(z0)) → c12(INF(z0)) 219.56/71.99
INF(ok(z0)) → c13(INF(z0)) 219.56/71.99
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/71.99
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/71.99
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/71.99
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/71.99
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/71.99
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/71.99
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/71.99
S(ok(z0)) → c30(S(z0)) 219.56/71.99
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/71.99
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/71.99
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/71.99
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c11(LENGTH(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c11(LENGTH(mark(false)), ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c11(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/71.99
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/71.99
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11

219.56/71.99

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

Split RHS of tuples not part of any SCC

219.56/71.99

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.56/71.99
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.56/71.99
active(eq(z0, z1)) → mark(false) 219.56/71.99
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.56/71.99
active(take(0, z0)) → mark(nil) 219.56/71.99
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.56/71.99
active(length(nil)) → mark(0) 219.56/71.99
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.56/71.99
active(inf(z0)) → inf(active(z0)) 219.56/71.99
active(take(z0, z1)) → take(active(z0), z1) 219.56/71.99
active(take(z0, z1)) → take(z0, active(z1)) 219.56/71.99
active(length(z0)) → length(active(z0)) 219.56/71.99
inf(mark(z0)) → mark(inf(z0)) 219.56/71.99
inf(ok(z0)) → ok(inf(z0)) 219.56/71.99
take(mark(z0), z1) → mark(take(z0, z1)) 219.56/71.99
take(z0, mark(z1)) → mark(take(z0, z1)) 219.56/71.99
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.56/71.99
length(mark(z0)) → mark(length(z0)) 219.56/71.99
length(ok(z0)) → ok(length(z0)) 219.56/71.99
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.56/71.99
proper(0) → ok(0) 219.56/71.99
proper(true) → ok(true) 219.56/71.99
proper(s(z0)) → s(proper(z0)) 219.56/71.99
proper(false) → ok(false) 219.56/71.99
proper(inf(z0)) → inf(proper(z0)) 219.56/71.99
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.56/71.99
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.56/71.99
proper(nil) → ok(nil) 219.56/71.99
proper(length(z0)) → length(proper(z0)) 219.56/71.99
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.56/71.99
s(ok(z0)) → ok(s(z0)) 219.56/71.99
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.56/71.99
top(mark(z0)) → top(proper(z0)) 219.56/71.99
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/71.99
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/71.99
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/71.99
INF(mark(z0)) → c12(INF(z0)) 219.56/71.99
INF(ok(z0)) → c13(INF(z0)) 219.56/71.99
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/71.99
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/71.99
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/71.99
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/71.99
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/71.99
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/71.99
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/71.99
S(ok(z0)) → c30(S(z0)) 219.56/71.99
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/71.99
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/71.99
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/71.99
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/71.99
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/71.99
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/71.99
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/71.99
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/71.99
INF(mark(z0)) → c12(INF(z0)) 219.56/71.99
INF(ok(z0)) → c13(INF(z0)) 219.56/71.99
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/71.99
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/71.99
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/71.99
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/71.99
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/71.99
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/71.99
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/71.99
S(ok(z0)) → c30(S(z0)) 219.56/71.99
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/71.99
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/71.99
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/71.99
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/71.99
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/71.99
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4

219.56/71.99

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

Use narrowing to replace PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 219.56/71.99
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 219.56/71.99
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.56/71.99
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 219.56/71.99
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.56/71.99
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.56/71.99
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 219.56/71.99
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.56/71.99
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 219.56/71.99
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 219.56/71.99
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 219.56/71.99
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 219.56/71.99
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

219.56/71.99

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.56/71.99
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.56/71.99
active(eq(z0, z1)) → mark(false) 219.56/71.99
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.56/71.99
active(take(0, z0)) → mark(nil) 219.56/71.99
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.56/71.99
active(length(nil)) → mark(0) 219.56/71.99
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.56/71.99
active(inf(z0)) → inf(active(z0)) 219.56/71.99
active(take(z0, z1)) → take(active(z0), z1) 219.56/71.99
active(take(z0, z1)) → take(z0, active(z1)) 219.56/71.99
active(length(z0)) → length(active(z0)) 219.56/71.99
inf(mark(z0)) → mark(inf(z0)) 219.56/71.99
inf(ok(z0)) → ok(inf(z0)) 219.56/71.99
take(mark(z0), z1) → mark(take(z0, z1)) 219.56/71.99
take(z0, mark(z1)) → mark(take(z0, z1)) 219.56/71.99
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.56/71.99
length(mark(z0)) → mark(length(z0)) 219.56/71.99
length(ok(z0)) → ok(length(z0)) 219.56/71.99
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.56/71.99
proper(0) → ok(0) 219.56/71.99
proper(true) → ok(true) 219.56/71.99
proper(s(z0)) → s(proper(z0)) 219.56/71.99
proper(false) → ok(false) 219.56/71.99
proper(inf(z0)) → inf(proper(z0)) 219.56/71.99
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.56/71.99
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.56/71.99
proper(nil) → ok(nil) 219.56/71.99
proper(length(z0)) → length(proper(z0)) 219.56/71.99
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.56/71.99
s(ok(z0)) → ok(s(z0)) 219.56/71.99
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.56/71.99
top(mark(z0)) → top(proper(z0)) 219.56/71.99
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/71.99
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/71.99
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/71.99
INF(mark(z0)) → c12(INF(z0)) 219.56/71.99
INF(ok(z0)) → c13(INF(z0)) 219.56/71.99
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/71.99
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/71.99
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/71.99
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/71.99
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/71.99
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/71.99
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/71.99
S(ok(z0)) → c30(S(z0)) 219.56/71.99
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/71.99
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/71.99
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/71.99
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/71.99
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/71.99
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 219.56/71.99
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 219.56/71.99
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.56/71.99
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 219.56/71.99
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.56/71.99
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.56/71.99
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 219.56/71.99
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.56/71.99
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 219.56/71.99
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 219.56/71.99
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 219.56/71.99
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 219.56/71.99
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/71.99
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/71.99
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/71.99
INF(mark(z0)) → c12(INF(z0)) 219.56/71.99
INF(ok(z0)) → c13(INF(z0)) 219.56/71.99
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/71.99
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/71.99
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/71.99
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/71.99
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/71.99
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/71.99
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/71.99
S(ok(z0)) → c30(S(z0)) 219.56/71.99
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/71.99
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/71.99
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/71.99
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/71.99
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/71.99
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 219.56/71.99
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 219.56/71.99
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.56/71.99
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 219.56/71.99
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.56/71.99
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.56/71.99
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 219.56/71.99
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.56/71.99
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 219.56/71.99
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 219.56/71.99
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 219.56/71.99
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 219.56/71.99
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19

219.56/71.99

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

Removed 8 trailing tuple parts

219.56/71.99

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.56/71.99
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.56/71.99
active(eq(z0, z1)) → mark(false) 219.56/71.99
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.56/71.99
active(take(0, z0)) → mark(nil) 219.56/71.99
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.56/71.99
active(length(nil)) → mark(0) 219.56/71.99
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.56/71.99
active(inf(z0)) → inf(active(z0)) 219.56/71.99
active(take(z0, z1)) → take(active(z0), z1) 219.56/71.99
active(take(z0, z1)) → take(z0, active(z1)) 219.56/71.99
active(length(z0)) → length(active(z0)) 219.56/71.99
inf(mark(z0)) → mark(inf(z0)) 219.56/71.99
inf(ok(z0)) → ok(inf(z0)) 219.56/71.99
take(mark(z0), z1) → mark(take(z0, z1)) 219.56/71.99
take(z0, mark(z1)) → mark(take(z0, z1)) 219.56/71.99
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.56/71.99
length(mark(z0)) → mark(length(z0)) 219.56/71.99
length(ok(z0)) → ok(length(z0)) 219.56/71.99
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.56/71.99
proper(0) → ok(0) 219.56/71.99
proper(true) → ok(true) 219.56/71.99
proper(s(z0)) → s(proper(z0)) 219.56/71.99
proper(false) → ok(false) 219.56/71.99
proper(inf(z0)) → inf(proper(z0)) 219.56/71.99
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.56/71.99
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.56/71.99
proper(nil) → ok(nil) 219.56/71.99
proper(length(z0)) → length(proper(z0)) 219.56/71.99
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.56/71.99
s(ok(z0)) → ok(s(z0)) 219.56/71.99
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.56/71.99
top(mark(z0)) → top(proper(z0)) 219.56/71.99
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/71.99
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/71.99
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/71.99
INF(mark(z0)) → c12(INF(z0)) 219.56/71.99
INF(ok(z0)) → c13(INF(z0)) 219.56/71.99
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/71.99
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/71.99
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/71.99
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/71.99
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/71.99
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/71.99
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/71.99
S(ok(z0)) → c30(S(z0)) 219.56/71.99
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/71.99
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/71.99
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/71.99
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/71.99
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/71.99
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.56/71.99
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.56/71.99
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.56/71.99
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.56/71.99
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.56/71.99
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/71.99
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/71.99
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/71.99
INF(mark(z0)) → c12(INF(z0)) 219.56/71.99
INF(ok(z0)) → c13(INF(z0)) 219.56/71.99
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/71.99
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/71.99
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/71.99
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/71.99
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/71.99
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/71.99
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/71.99
S(ok(z0)) → c30(S(z0)) 219.56/71.99
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/71.99
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/71.99
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/71.99
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/71.99
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/71.99
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.56/71.99
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.56/71.99
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.56/71.99
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.56/71.99
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.56/71.99
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19

219.56/71.99

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

Use narrowing to replace PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) by

PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(s(0)) → c22(S(ok(0)), PROPER(0)) 219.56/71.99
PROPER(s(true)) → c22(S(ok(true)), PROPER(true)) 219.56/71.99
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.56/71.99
PROPER(s(false)) → c22(S(ok(false)), PROPER(false)) 219.56/71.99
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.56/71.99
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.56/71.99
PROPER(s(nil)) → c22(S(ok(nil)), PROPER(nil)) 219.56/71.99
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))

219.56/71.99

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.56/71.99
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.56/71.99
active(eq(z0, z1)) → mark(false) 219.56/71.99
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.56/71.99
active(take(0, z0)) → mark(nil) 219.56/71.99
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.56/71.99
active(length(nil)) → mark(0) 219.56/71.99
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.56/71.99
active(inf(z0)) → inf(active(z0)) 219.56/71.99
active(take(z0, z1)) → take(active(z0), z1) 219.56/71.99
active(take(z0, z1)) → take(z0, active(z1)) 219.56/71.99
active(length(z0)) → length(active(z0)) 219.56/71.99
inf(mark(z0)) → mark(inf(z0)) 219.56/71.99
inf(ok(z0)) → ok(inf(z0)) 219.56/71.99
take(mark(z0), z1) → mark(take(z0, z1)) 219.56/71.99
take(z0, mark(z1)) → mark(take(z0, z1)) 219.56/71.99
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.56/71.99
length(mark(z0)) → mark(length(z0)) 219.56/71.99
length(ok(z0)) → ok(length(z0)) 219.56/71.99
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.56/71.99
proper(0) → ok(0) 219.56/71.99
proper(true) → ok(true) 219.56/71.99
proper(s(z0)) → s(proper(z0)) 219.56/71.99
proper(false) → ok(false) 219.56/71.99
proper(inf(z0)) → inf(proper(z0)) 219.56/71.99
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.56/71.99
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.56/71.99
proper(nil) → ok(nil) 219.56/71.99
proper(length(z0)) → length(proper(z0)) 219.56/71.99
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.56/71.99
s(ok(z0)) → ok(s(z0)) 219.56/71.99
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.56/71.99
top(mark(z0)) → top(proper(z0)) 219.56/71.99
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/71.99
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/71.99
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/71.99
INF(mark(z0)) → c12(INF(z0)) 219.56/71.99
INF(ok(z0)) → c13(INF(z0)) 219.56/71.99
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/71.99
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/71.99
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/71.99
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/71.99
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/71.99
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/71.99
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/71.99
S(ok(z0)) → c30(S(z0)) 219.56/71.99
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/71.99
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/71.99
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/71.99
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/71.99
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/71.99
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.56/71.99
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.56/71.99
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.56/71.99
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.56/71.99
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.56/71.99
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(s(0)) → c22(S(ok(0)), PROPER(0)) 219.56/71.99
PROPER(s(true)) → c22(S(ok(true)), PROPER(true)) 219.56/71.99
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.56/71.99
PROPER(s(false)) → c22(S(ok(false)), PROPER(false)) 219.56/71.99
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.56/71.99
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.56/71.99
PROPER(s(nil)) → c22(S(ok(nil)), PROPER(nil)) 219.56/71.99
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/71.99
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/71.99
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/71.99
INF(mark(z0)) → c12(INF(z0)) 219.56/71.99
INF(ok(z0)) → c13(INF(z0)) 219.56/71.99
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/71.99
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/71.99
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/71.99
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/71.99
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/71.99
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/71.99
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/71.99
S(ok(z0)) → c30(S(z0)) 219.56/71.99
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/71.99
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/71.99
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/71.99
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/71.99
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/71.99
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.56/71.99
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.56/71.99
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.56/71.99
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.56/71.99
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.56/71.99
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(s(0)) → c22(S(ok(0)), PROPER(0)) 219.56/71.99
PROPER(s(true)) → c22(S(ok(true)), PROPER(true)) 219.56/71.99
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.56/71.99
PROPER(s(false)) → c22(S(ok(false)), PROPER(false)) 219.56/71.99
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.56/71.99
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.56/71.99
PROPER(s(nil)) → c22(S(ok(nil)), PROPER(nil)) 219.56/71.99
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22

219.56/71.99

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

Removed 4 trailing tuple parts

219.56/71.99

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.56/71.99
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.56/71.99
active(eq(z0, z1)) → mark(false) 219.56/71.99
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.56/71.99
active(take(0, z0)) → mark(nil) 219.56/71.99
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.56/71.99
active(length(nil)) → mark(0) 219.56/71.99
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.56/71.99
active(inf(z0)) → inf(active(z0)) 219.56/71.99
active(take(z0, z1)) → take(active(z0), z1) 219.56/71.99
active(take(z0, z1)) → take(z0, active(z1)) 219.56/71.99
active(length(z0)) → length(active(z0)) 219.56/71.99
inf(mark(z0)) → mark(inf(z0)) 219.56/71.99
inf(ok(z0)) → ok(inf(z0)) 219.56/71.99
take(mark(z0), z1) → mark(take(z0, z1)) 219.56/71.99
take(z0, mark(z1)) → mark(take(z0, z1)) 219.56/71.99
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.56/71.99
length(mark(z0)) → mark(length(z0)) 219.56/71.99
length(ok(z0)) → ok(length(z0)) 219.56/71.99
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.56/71.99
proper(0) → ok(0) 219.56/71.99
proper(true) → ok(true) 219.56/71.99
proper(s(z0)) → s(proper(z0)) 219.56/71.99
proper(false) → ok(false) 219.56/71.99
proper(inf(z0)) → inf(proper(z0)) 219.56/71.99
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.56/71.99
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.56/71.99
proper(nil) → ok(nil) 219.56/71.99
proper(length(z0)) → length(proper(z0)) 219.56/71.99
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.56/71.99
s(ok(z0)) → ok(s(z0)) 219.56/71.99
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.56/71.99
top(mark(z0)) → top(proper(z0)) 219.56/71.99
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/71.99
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/71.99
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/71.99
INF(mark(z0)) → c12(INF(z0)) 219.56/71.99
INF(ok(z0)) → c13(INF(z0)) 219.56/71.99
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/71.99
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/71.99
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/71.99
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/71.99
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/71.99
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/71.99
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/71.99
S(ok(z0)) → c30(S(z0)) 219.56/71.99
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/71.99
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/71.99
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/71.99
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/71.99
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/71.99
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.56/71.99
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.56/71.99
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.56/71.99
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.56/71.99
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.56/71.99
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.56/71.99
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.56/71.99
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.56/71.99
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.56/71.99
PROPER(s(0)) → c22(S(ok(0))) 219.56/71.99
PROPER(s(true)) → c22(S(ok(true))) 219.56/71.99
PROPER(s(false)) → c22(S(ok(false))) 219.56/71.99
PROPER(s(nil)) → c22(S(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/71.99
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/71.99
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/71.99
INF(mark(z0)) → c12(INF(z0)) 219.56/71.99
INF(ok(z0)) → c13(INF(z0)) 219.56/71.99
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/71.99
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/71.99
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/71.99
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/71.99
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/71.99
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) 219.56/71.99
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/71.99
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/71.99
S(ok(z0)) → c30(S(z0)) 219.56/71.99
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/71.99
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/71.99
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/71.99
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/71.99
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/71.99
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.56/71.99
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.56/71.99
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.56/71.99
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.56/71.99
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.56/71.99
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.56/71.99
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.56/71.99
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.56/71.99
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.56/71.99
PROPER(s(0)) → c22(S(ok(0))) 219.56/71.99
PROPER(s(true)) → c22(S(ok(true))) 219.56/71.99
PROPER(s(false)) → c22(S(ok(false))) 219.56/71.99
PROPER(s(nil)) → c22(S(ok(nil)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22

219.56/71.99

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

Use narrowing to replace PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) by

PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(inf(0)) → c24(INF(ok(0)), PROPER(0)) 219.56/71.99
PROPER(inf(true)) → c24(INF(ok(true)), PROPER(true)) 219.56/71.99
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.56/71.99
PROPER(inf(false)) → c24(INF(ok(false)), PROPER(false)) 219.56/71.99
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.56/71.99
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.56/71.99
PROPER(inf(nil)) → c24(INF(ok(nil)), PROPER(nil)) 219.56/71.99
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))

219.56/71.99

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.56/71.99
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.56/71.99
active(eq(z0, z1)) → mark(false) 219.56/71.99
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.56/71.99
active(take(0, z0)) → mark(nil) 219.56/71.99
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.56/71.99
active(length(nil)) → mark(0) 219.56/71.99
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.56/71.99
active(inf(z0)) → inf(active(z0)) 219.56/71.99
active(take(z0, z1)) → take(active(z0), z1) 219.56/71.99
active(take(z0, z1)) → take(z0, active(z1)) 219.56/71.99
active(length(z0)) → length(active(z0)) 219.56/71.99
inf(mark(z0)) → mark(inf(z0)) 219.56/71.99
inf(ok(z0)) → ok(inf(z0)) 219.56/71.99
take(mark(z0), z1) → mark(take(z0, z1)) 219.56/71.99
take(z0, mark(z1)) → mark(take(z0, z1)) 219.56/71.99
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.56/71.99
length(mark(z0)) → mark(length(z0)) 219.56/71.99
length(ok(z0)) → ok(length(z0)) 219.56/71.99
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.56/71.99
proper(0) → ok(0) 219.56/71.99
proper(true) → ok(true) 219.56/71.99
proper(s(z0)) → s(proper(z0)) 219.56/71.99
proper(false) → ok(false) 219.56/71.99
proper(inf(z0)) → inf(proper(z0)) 219.56/71.99
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.56/71.99
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.56/71.99
proper(nil) → ok(nil) 219.56/71.99
proper(length(z0)) → length(proper(z0)) 219.56/71.99
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.56/71.99
s(ok(z0)) → ok(s(z0)) 219.56/71.99
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.56/71.99
top(mark(z0)) → top(proper(z0)) 219.56/71.99
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/71.99
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/71.99
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/71.99
INF(mark(z0)) → c12(INF(z0)) 219.56/71.99
INF(ok(z0)) → c13(INF(z0)) 219.56/71.99
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/71.99
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/71.99
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/71.99
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/71.99
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/71.99
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/71.99
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/71.99
S(ok(z0)) → c30(S(z0)) 219.56/71.99
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/71.99
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/71.99
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/71.99
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/71.99
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/71.99
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/71.99
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/71.99
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.56/71.99
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.56/71.99
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.56/71.99
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.56/71.99
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/71.99
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.56/71.99
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.56/71.99
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.56/71.99
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.56/71.99
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.56/71.99
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.56/71.99
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.56/71.99
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.56/71.99
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.56/71.99
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.56/71.99
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.56/71.99
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.56/71.99
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.56/71.99
PROPER(s(0)) → c22(S(ok(0))) 219.56/71.99
PROPER(s(true)) → c22(S(ok(true))) 219.56/71.99
PROPER(s(false)) → c22(S(ok(false))) 219.56/71.99
PROPER(s(nil)) → c22(S(ok(nil))) 219.56/71.99
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.56/71.99
PROPER(inf(0)) → c24(INF(ok(0)), PROPER(0)) 219.56/71.99
PROPER(inf(true)) → c24(INF(ok(true)), PROPER(true)) 219.56/71.99
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.56/71.99
PROPER(inf(false)) → c24(INF(ok(false)), PROPER(false)) 219.56/71.99
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.56/71.99
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.56/71.99
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.56/71.99
PROPER(inf(nil)) → c24(INF(ok(nil)), PROPER(nil)) 219.56/71.99
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/71.99
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/71.99
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/71.99
INF(mark(z0)) → c12(INF(z0)) 219.56/71.99
INF(ok(z0)) → c13(INF(z0)) 219.56/71.99
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/71.99
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/71.99
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/71.99
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/71.99
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/71.99
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/71.99
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/71.99
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/71.99
S(ok(z0)) → c30(S(z0)) 219.56/71.99
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/71.99
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/71.99
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/71.99
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/71.99
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/71.99
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/71.99
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/71.99
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/71.99
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/72.00
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/72.00
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/72.00
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/72.00
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/72.00
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/72.00
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/72.00
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/72.00
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/72.00
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/72.00
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/72.00
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/72.00
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/72.00
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/72.00
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/72.00
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/72.00
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.56/72.00
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.56/72.00
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/72.00
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.56/72.00
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.56/72.00
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.56/72.00
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.56/72.00
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.56/72.00
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/72.00
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.56/72.00
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.56/72.00
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.56/72.00
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.56/72.00
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.56/72.00
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.56/72.00
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.56/72.00
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.56/72.00
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.56/72.00
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.56/72.00
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.56/72.00
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.56/72.00
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.56/72.00
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.56/72.00
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.56/72.00
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.56/72.00
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.56/72.00
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.56/72.00
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.56/72.00
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.56/72.00
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.56/72.00
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.56/72.00
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.56/72.00
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.56/72.00
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.56/72.00
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.56/72.00
PROPER(s(0)) → c22(S(ok(0))) 219.56/72.00
PROPER(s(true)) → c22(S(ok(true))) 219.56/72.00
PROPER(s(false)) → c22(S(ok(false))) 219.56/72.00
PROPER(s(nil)) → c22(S(ok(nil))) 219.56/72.00
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.56/72.00
PROPER(inf(0)) → c24(INF(ok(0)), PROPER(0)) 219.56/72.00
PROPER(inf(true)) → c24(INF(ok(true)), PROPER(true)) 219.56/72.00
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.56/72.00
PROPER(inf(false)) → c24(INF(ok(false)), PROPER(false)) 219.56/72.00
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.56/72.00
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.56/72.00
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.56/72.00
PROPER(inf(nil)) → c24(INF(ok(nil)), PROPER(nil)) 219.56/72.00
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24

219.56/72.00

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

Removed 4 trailing tuple parts

219.56/72.00

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.56/72.00
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.56/72.00
active(eq(z0, z1)) → mark(false) 219.56/72.00
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.56/72.00
active(take(0, z0)) → mark(nil) 219.56/72.00
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.56/72.00
active(length(nil)) → mark(0) 219.56/72.00
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.56/72.00
active(inf(z0)) → inf(active(z0)) 219.56/72.00
active(take(z0, z1)) → take(active(z0), z1) 219.56/72.00
active(take(z0, z1)) → take(z0, active(z1)) 219.56/72.00
active(length(z0)) → length(active(z0)) 219.56/72.00
inf(mark(z0)) → mark(inf(z0)) 219.56/72.00
inf(ok(z0)) → ok(inf(z0)) 219.56/72.00
take(mark(z0), z1) → mark(take(z0, z1)) 219.56/72.00
take(z0, mark(z1)) → mark(take(z0, z1)) 219.56/72.00
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.56/72.00
length(mark(z0)) → mark(length(z0)) 219.56/72.00
length(ok(z0)) → ok(length(z0)) 219.56/72.00
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.56/72.00
proper(0) → ok(0) 219.56/72.00
proper(true) → ok(true) 219.56/72.00
proper(s(z0)) → s(proper(z0)) 219.56/72.00
proper(false) → ok(false) 219.56/72.00
proper(inf(z0)) → inf(proper(z0)) 219.56/72.00
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.56/72.00
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.56/72.00
proper(nil) → ok(nil) 219.56/72.00
proper(length(z0)) → length(proper(z0)) 219.56/72.00
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.56/72.00
s(ok(z0)) → ok(s(z0)) 219.56/72.00
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.56/72.00
top(mark(z0)) → top(proper(z0)) 219.56/72.00
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.56/72.00
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.56/72.00
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.56/72.00
INF(mark(z0)) → c12(INF(z0)) 219.56/72.00
INF(ok(z0)) → c13(INF(z0)) 219.56/72.00
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.56/72.00
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.56/72.00
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.56/72.00
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.56/72.00
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.56/72.00
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/72.00
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.56/72.00
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.56/72.00
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.56/72.00
S(ok(z0)) → c30(S(z0)) 219.56/72.00
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.56/72.00
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.56/72.00
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.56/72.00
ACTIVE(inf(z0)) → c3(S(z0)) 219.56/72.00
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.56/72.00
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.56/72.00
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.56/72.00
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.56/72.00
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/72.00
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.56/72.00
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/72.00
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.56/72.00
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/72.00
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/72.00
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.56/72.00
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.56/72.00
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.56/72.00
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.56/72.00
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.56/72.00
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.56/72.00
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.56/72.00
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.56/72.00
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.56/72.00
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.56/72.00
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.56/72.00
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.56/72.00
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.56/72.00
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.56/72.00
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.56/72.00
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.56/72.00
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.56/72.00
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/72.00
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.56/72.00
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.56/72.00
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.56/72.00
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.56/72.00
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.56/72.00
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.56/72.00
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.56/72.00
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.56/72.00
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.56/72.00
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.56/72.00
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.56/72.00
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.56/72.00
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.56/72.00
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.56/72.00
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.56/72.00
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.56/72.00
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.56/72.00
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.56/72.00
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.00
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.00
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.00
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.00
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.00
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.00
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.00
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.00
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.00
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.00
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.00
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.00
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.00
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.00
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.00
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.00
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.00
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.00
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.00
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.00
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.00
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.00
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.00
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.00
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.00
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.00
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.00
PROPER(inf(nil)) → c24(INF(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.00
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.00
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.00
INF(mark(z0)) → c12(INF(z0)) 219.93/72.00
INF(ok(z0)) → c13(INF(z0)) 219.93/72.00
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.00
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.00
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.00
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.00
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.00
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.93/72.00
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.93/72.00
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.93/72.00
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.00
S(ok(z0)) → c30(S(z0)) 219.93/72.00
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.00
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.93/72.00
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.00
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.00
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.00
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.00
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.00
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.00
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.00
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.00
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.00
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.00
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.00
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.00
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.00
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.00
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.00
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.00
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.00
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.00
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.00
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.00
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.00
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.00
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.00
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.00
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.00
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.00
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.00
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.00
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.00
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.00
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.00
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.00
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.00
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.00
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.00
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.00
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.00
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.00
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.00
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.00
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.00
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.00
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.00
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.00
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.00
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.00
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.00
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.00
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.00
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.00
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.00
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.00
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.00
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.00
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.00
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.00
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.00
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.00
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.00
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.00
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.00
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.00
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.00
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.00
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.00
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.00
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.00
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.00
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.00
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.00
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.00
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.00
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.00
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.00
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.00
PROPER(inf(nil)) → c24(INF(ok(nil)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24

219.93/72.00

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

Use narrowing to replace PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.00
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 219.93/72.00
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 219.93/72.00
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.00
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 219.93/72.00
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.00
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.00
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.00
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 219.93/72.00
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.00
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.00
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 219.93/72.00
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 219.93/72.00
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.00
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 219.93/72.00
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.00
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.00
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.00
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 219.93/72.00
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

219.93/72.01

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.01
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.01
active(eq(z0, z1)) → mark(false) 219.93/72.01
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.01
active(take(0, z0)) → mark(nil) 219.93/72.01
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.01
active(length(nil)) → mark(0) 219.93/72.01
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.01
active(inf(z0)) → inf(active(z0)) 219.93/72.01
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.01
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.01
active(length(z0)) → length(active(z0)) 219.93/72.01
inf(mark(z0)) → mark(inf(z0)) 219.93/72.01
inf(ok(z0)) → ok(inf(z0)) 219.93/72.01
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.01
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.01
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.01
length(mark(z0)) → mark(length(z0)) 219.93/72.01
length(ok(z0)) → ok(length(z0)) 219.93/72.01
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.01
proper(0) → ok(0) 219.93/72.01
proper(true) → ok(true) 219.93/72.01
proper(s(z0)) → s(proper(z0)) 219.93/72.01
proper(false) → ok(false) 219.93/72.01
proper(inf(z0)) → inf(proper(z0)) 219.93/72.01
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.01
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.01
proper(nil) → ok(nil) 219.93/72.01
proper(length(z0)) → length(proper(z0)) 219.93/72.01
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.01
s(ok(z0)) → ok(s(z0)) 219.93/72.01
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.01
top(mark(z0)) → top(proper(z0)) 219.93/72.01
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.01
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.01
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.01
INF(mark(z0)) → c12(INF(z0)) 219.93/72.01
INF(ok(z0)) → c13(INF(z0)) 219.93/72.01
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.01
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.01
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.01
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.01
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.01
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.93/72.01
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.93/72.01
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.01
S(ok(z0)) → c30(S(z0)) 219.93/72.01
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.01
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.93/72.01
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.01
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.01
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.01
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.01
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.01
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.01
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.01
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.01
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.01
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.01
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.01
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.01
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.01
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.01
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.01
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.01
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.01
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.01
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.01
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.01
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.01
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.01
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.01
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.01
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.01
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.01
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.01
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.01
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.01
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.01
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.01
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.01
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.01
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.01
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.01
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.01
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.01
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.01
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.01
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.01
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.01
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.01
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.01
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.01
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.01
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.01
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.01
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.01
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.01
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 219.93/72.01
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 219.93/72.01
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.01
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 219.93/72.01
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.01
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.01
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 219.93/72.01
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.01
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 219.93/72.01
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 219.93/72.01
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.01
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 219.93/72.01
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.01
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 219.93/72.01
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.01
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.01
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.01
INF(mark(z0)) → c12(INF(z0)) 219.93/72.01
INF(ok(z0)) → c13(INF(z0)) 219.93/72.01
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.01
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.01
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.01
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.01
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.01
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.93/72.01
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.93/72.01
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.01
S(ok(z0)) → c30(S(z0)) 219.93/72.01
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.01
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.93/72.01
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.01
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.01
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.01
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.01
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.01
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.01
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.01
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.01
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.01
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.01
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.01
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.01
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.01
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.01
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.01
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.01
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.01
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.01
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.01
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.01
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.01
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.01
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.01
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.01
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.01
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.01
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.01
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.01
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.01
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.01
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.01
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.01
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.01
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.01
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.01
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.01
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.01
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.01
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.01
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.01
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.01
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.01
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.01
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.01
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.01
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.01
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.01
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.01
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.01
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 219.93/72.01
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 219.93/72.01
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.01
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 219.93/72.01
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.01
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.01
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 219.93/72.01
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.01
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 219.93/72.01
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 219.93/72.01
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.01
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 219.93/72.01
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.01
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 219.93/72.01
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25

219.93/72.01

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

Removed 8 trailing tuple parts

219.93/72.01

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.01
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.01
active(eq(z0, z1)) → mark(false) 219.93/72.01
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.01
active(take(0, z0)) → mark(nil) 219.93/72.01
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.01
active(length(nil)) → mark(0) 219.93/72.01
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.01
active(inf(z0)) → inf(active(z0)) 219.93/72.01
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.01
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.01
active(length(z0)) → length(active(z0)) 219.93/72.01
inf(mark(z0)) → mark(inf(z0)) 219.93/72.01
inf(ok(z0)) → ok(inf(z0)) 219.93/72.01
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.01
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.01
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.01
length(mark(z0)) → mark(length(z0)) 219.93/72.01
length(ok(z0)) → ok(length(z0)) 219.93/72.01
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.01
proper(0) → ok(0) 219.93/72.01
proper(true) → ok(true) 219.93/72.01
proper(s(z0)) → s(proper(z0)) 219.93/72.01
proper(false) → ok(false) 219.93/72.01
proper(inf(z0)) → inf(proper(z0)) 219.93/72.01
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.01
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.01
proper(nil) → ok(nil) 219.93/72.01
proper(length(z0)) → length(proper(z0)) 219.93/72.01
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.01
s(ok(z0)) → ok(s(z0)) 219.93/72.01
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.01
top(mark(z0)) → top(proper(z0)) 219.93/72.01
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.01
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.01
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.01
INF(mark(z0)) → c12(INF(z0)) 219.93/72.01
INF(ok(z0)) → c13(INF(z0)) 219.93/72.01
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.01
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.01
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.01
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.01
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.01
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.93/72.01
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.93/72.01
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.01
S(ok(z0)) → c30(S(z0)) 219.93/72.01
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.01
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.93/72.01
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.01
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.01
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.01
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.01
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.01
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.01
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.01
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.01
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.01
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.01
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.01
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.01
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.01
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.01
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.01
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.01
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.01
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.01
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.01
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.01
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.01
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.01
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.01
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.01
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.01
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.01
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.01
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.01
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.01
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.01
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.01
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.01
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.01
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.01
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.01
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.01
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.01
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.01
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.01
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.01
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.01
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.01
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.01
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.01
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.01
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.01
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.01
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.01
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.01
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.01
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.01
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.01
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.01
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.01
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.01
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.01
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.01
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.01
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.01
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.01
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.01
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.01
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.01
INF(mark(z0)) → c12(INF(z0)) 219.93/72.01
INF(ok(z0)) → c13(INF(z0)) 219.93/72.01
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.01
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.01
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.01
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.01
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.01
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 219.93/72.01
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.93/72.01
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.01
S(ok(z0)) → c30(S(z0)) 219.93/72.01
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.01
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.93/72.01
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.01
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.01
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.01
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.01
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.01
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.01
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.01
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.01
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.01
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.01
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.01
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.01
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.01
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.01
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.01
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.01
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.01
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.01
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.01
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.01
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.01
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.01
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.01
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.01
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.01
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.01
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.01
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.01
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.01
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.01
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.01
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.01
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.01
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.01
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.01
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.01
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.01
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.01
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.01
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.01
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.01
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.01
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.01
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.01
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.01
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.01
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.01
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.01
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.01
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.01
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.01
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.01
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.01
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.01
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.01
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.01
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.01
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.01
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.01
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.01
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25

219.93/72.01

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

Use narrowing to replace PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 219.93/72.01
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 219.93/72.01
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.01
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 219.93/72.01
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.01
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.01
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 219.93/72.01
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.01
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 219.93/72.01
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 219.93/72.01
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.01
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 219.93/72.01
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.01
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 219.93/72.01
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

219.93/72.01

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.01
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.01
active(eq(z0, z1)) → mark(false) 219.93/72.01
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.01
active(take(0, z0)) → mark(nil) 219.93/72.01
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.01
active(length(nil)) → mark(0) 219.93/72.01
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.01
active(inf(z0)) → inf(active(z0)) 219.93/72.01
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.01
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.01
active(length(z0)) → length(active(z0)) 219.93/72.01
inf(mark(z0)) → mark(inf(z0)) 219.93/72.01
inf(ok(z0)) → ok(inf(z0)) 219.93/72.01
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.01
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.01
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.01
length(mark(z0)) → mark(length(z0)) 219.93/72.01
length(ok(z0)) → ok(length(z0)) 219.93/72.01
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.01
proper(0) → ok(0) 219.93/72.01
proper(true) → ok(true) 219.93/72.01
proper(s(z0)) → s(proper(z0)) 219.93/72.01
proper(false) → ok(false) 219.93/72.01
proper(inf(z0)) → inf(proper(z0)) 219.93/72.01
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.01
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.01
proper(nil) → ok(nil) 219.93/72.01
proper(length(z0)) → length(proper(z0)) 219.93/72.01
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.01
s(ok(z0)) → ok(s(z0)) 219.93/72.01
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.01
top(mark(z0)) → top(proper(z0)) 219.93/72.01
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.01
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.01
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.01
INF(mark(z0)) → c12(INF(z0)) 219.93/72.01
INF(ok(z0)) → c13(INF(z0)) 219.93/72.01
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.01
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.01
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.01
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.01
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.01
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.93/72.01
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.01
S(ok(z0)) → c30(S(z0)) 219.93/72.01
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.01
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.93/72.01
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.01
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.01
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.01
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.01
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.01
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.01
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.01
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.01
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.01
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.01
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.01
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.01
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.01
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.01
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.01
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.01
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.01
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.01
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.01
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.01
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.01
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.01
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.01
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.01
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.01
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.01
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.01
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.01
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.01
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.01
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.01
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.01
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.01
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.01
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.01
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.01
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.01
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.01
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.01
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.01
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.01
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.02
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.02
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.02
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.02
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.02
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.02
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.02
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.02
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.02
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.02
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.02
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.02
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.02
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.02
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.02
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.02
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.02
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.02
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.02
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.02
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.02
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.02
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.02
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.02
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.02
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.02
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.02
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.02
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.02
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.02
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.02
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.02
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.02
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.02
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.02
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.02
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.02
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.02
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.02
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.02
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.02
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.02
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 219.93/72.02
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 219.93/72.02
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.02
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 219.93/72.02
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.02
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.02
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.02
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 219.93/72.02
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.02
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.02
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 219.93/72.02
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 219.93/72.02
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.02
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 219.93/72.02
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.02
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.02
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.02
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 219.93/72.02
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.02
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.02
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.02
INF(mark(z0)) → c12(INF(z0)) 219.93/72.02
INF(ok(z0)) → c13(INF(z0)) 219.93/72.02
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.02
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.02
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.02
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.02
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.02
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.93/72.02
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.02
S(ok(z0)) → c30(S(z0)) 219.93/72.02
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.02
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.93/72.02
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.02
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.02
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.02
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.02
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.02
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.02
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.02
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.02
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.02
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.02
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.02
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.02
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.02
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.02
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.02
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.02
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.02
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.02
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.02
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.02
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.02
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.02
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.02
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.02
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.02
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.02
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.02
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.02
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.02
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.02
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.02
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.02
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.02
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.02
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.02
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.02
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.02
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.02
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.02
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.02
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.02
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.02
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.02
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.02
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.02
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.02
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.02
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.02
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.02
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.02
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.02
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.02
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.02
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.02
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.02
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.02
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.02
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.02
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.02
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.02
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.02
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.02
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.02
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.02
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.02
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.02
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.02
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.02
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.02
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.02
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.02
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.02
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.02
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.02
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.02
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.02
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.02
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.02
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.02
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.02
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.02
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.02
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.02
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.02
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.02
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.02
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.02
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.02
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.02
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.02
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.02
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.02
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.02
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.02
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.02
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 219.93/72.03
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 219.93/72.03
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.03
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 219.93/72.03
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.03
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.03
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 219.93/72.03
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.03
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 219.93/72.03
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 219.93/72.03
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.03
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 219.93/72.03
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.03
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 219.93/72.03
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26

219.93/72.03

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

Removed 8 trailing tuple parts

219.93/72.03

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.03
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.03
active(eq(z0, z1)) → mark(false) 219.93/72.03
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.03
active(take(0, z0)) → mark(nil) 219.93/72.03
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.03
active(length(nil)) → mark(0) 219.93/72.03
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.03
active(inf(z0)) → inf(active(z0)) 219.93/72.03
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.03
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.03
active(length(z0)) → length(active(z0)) 219.93/72.03
inf(mark(z0)) → mark(inf(z0)) 219.93/72.03
inf(ok(z0)) → ok(inf(z0)) 219.93/72.03
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.03
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.03
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.03
length(mark(z0)) → mark(length(z0)) 219.93/72.03
length(ok(z0)) → ok(length(z0)) 219.93/72.03
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.03
proper(0) → ok(0) 219.93/72.03
proper(true) → ok(true) 219.93/72.03
proper(s(z0)) → s(proper(z0)) 219.93/72.03
proper(false) → ok(false) 219.93/72.03
proper(inf(z0)) → inf(proper(z0)) 219.93/72.03
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.03
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.03
proper(nil) → ok(nil) 219.93/72.03
proper(length(z0)) → length(proper(z0)) 219.93/72.03
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.03
s(ok(z0)) → ok(s(z0)) 219.93/72.03
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.03
top(mark(z0)) → top(proper(z0)) 219.93/72.03
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.03
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.03
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.03
INF(mark(z0)) → c12(INF(z0)) 219.93/72.03
INF(ok(z0)) → c13(INF(z0)) 219.93/72.03
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.03
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.03
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.03
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.03
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.03
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.93/72.03
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.03
S(ok(z0)) → c30(S(z0)) 219.93/72.03
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.03
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.93/72.03
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.03
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.03
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.03
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.03
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.03
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.03
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.03
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.03
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.03
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.03
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.03
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.03
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.03
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.03
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.03
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.03
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.03
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.03
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.03
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.03
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.03
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.03
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.03
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.03
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.03
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.03
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.03
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.03
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.03
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.03
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.03
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.03
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.03
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.03
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.03
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.03
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.03
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.03
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.03
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.03
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.03
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.03
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.03
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.03
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.03
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.03
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.03
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.03
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.03
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.03
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.03
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.03
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.03
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.03
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.03
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.03
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.03
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.03
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.03
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.03
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.03
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.03
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.03
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.03
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.03
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.03
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.03
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.03
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.03
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.03
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.03
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.03
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.03
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.03
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.03
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.03
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.03
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.03
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.03
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.03
INF(mark(z0)) → c12(INF(z0)) 219.93/72.03
INF(ok(z0)) → c13(INF(z0)) 219.93/72.03
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.03
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.03
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.03
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.03
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.03
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) 219.93/72.03
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.03
S(ok(z0)) → c30(S(z0)) 219.93/72.03
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.03
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.93/72.03
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.03
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.03
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.03
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.03
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.03
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.03
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.03
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.03
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.03
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.03
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.03
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.03
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.03
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.03
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.03
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.03
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.03
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.03
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.03
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.03
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.03
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.03
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.03
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.03
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.03
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.03
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.03
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.03
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.03
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.03
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.03
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.03
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.03
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.03
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.03
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.03
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.03
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.03
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.03
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.03
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.03
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.03
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.03
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.03
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.03
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.03
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.03
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.03
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.03
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.03
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.03
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.03
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.03
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.03
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.03
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.03
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.03
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.03
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.03
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.03
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.03
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.03
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.03
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.03
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.03
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.03
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.03
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.03
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.03
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.03
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.03
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.03
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.03
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.03
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.03
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.03
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.03
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26

219.93/72.03

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

Use narrowing to replace PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) by

PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(length(0)) → c28(LENGTH(ok(0)), PROPER(0)) 219.93/72.03
PROPER(length(true)) → c28(LENGTH(ok(true)), PROPER(true)) 219.93/72.03
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.03
PROPER(length(false)) → c28(LENGTH(ok(false)), PROPER(false)) 219.93/72.03
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.03
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.03
PROPER(length(nil)) → c28(LENGTH(ok(nil)), PROPER(nil)) 219.93/72.03
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))

219.93/72.03

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.03
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.03
active(eq(z0, z1)) → mark(false) 219.93/72.03
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.03
active(take(0, z0)) → mark(nil) 219.93/72.03
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.03
active(length(nil)) → mark(0) 219.93/72.03
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.03
active(inf(z0)) → inf(active(z0)) 219.93/72.03
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.03
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.03
active(length(z0)) → length(active(z0)) 219.93/72.03
inf(mark(z0)) → mark(inf(z0)) 219.93/72.03
inf(ok(z0)) → ok(inf(z0)) 219.93/72.03
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.03
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.03
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.03
length(mark(z0)) → mark(length(z0)) 219.93/72.03
length(ok(z0)) → ok(length(z0)) 219.93/72.03
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.03
proper(0) → ok(0) 219.93/72.03
proper(true) → ok(true) 219.93/72.03
proper(s(z0)) → s(proper(z0)) 219.93/72.03
proper(false) → ok(false) 219.93/72.03
proper(inf(z0)) → inf(proper(z0)) 219.93/72.03
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.03
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.03
proper(nil) → ok(nil) 219.93/72.03
proper(length(z0)) → length(proper(z0)) 219.93/72.03
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.03
s(ok(z0)) → ok(s(z0)) 219.93/72.03
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.03
top(mark(z0)) → top(proper(z0)) 219.93/72.03
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.03
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.03
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.03
INF(mark(z0)) → c12(INF(z0)) 219.93/72.03
INF(ok(z0)) → c13(INF(z0)) 219.93/72.03
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.03
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.03
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.03
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.03
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.03
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.03
S(ok(z0)) → c30(S(z0)) 219.93/72.03
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.03
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.93/72.03
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.03
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.03
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.03
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.03
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.03
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.03
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.03
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.03
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.03
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.03
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.03
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.03
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.03
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.03
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.03
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.03
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.03
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.03
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.03
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.03
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.03
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.03
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.03
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.03
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.03
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.03
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.03
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.03
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.03
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.03
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.03
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.03
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.03
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.03
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.03
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.03
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.03
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.03
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.03
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.03
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.03
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.03
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.03
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.03
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.03
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.03
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.03
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.03
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.03
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.03
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.03
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.03
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.03
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.03
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.03
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.03
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.03
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.03
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.03
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.03
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.03
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.03
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.03
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.03
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.03
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.03
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.03
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.03
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.03
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.03
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.03
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.03
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.03
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.03
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.03
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.03
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.03
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(length(0)) → c28(LENGTH(ok(0)), PROPER(0)) 219.93/72.03
PROPER(length(true)) → c28(LENGTH(ok(true)), PROPER(true)) 219.93/72.03
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.03
PROPER(length(false)) → c28(LENGTH(ok(false)), PROPER(false)) 219.93/72.03
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.03
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.03
PROPER(length(nil)) → c28(LENGTH(ok(nil)), PROPER(nil)) 219.93/72.03
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.03
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.03
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.03
INF(mark(z0)) → c12(INF(z0)) 219.93/72.03
INF(ok(z0)) → c13(INF(z0)) 219.93/72.03
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.03
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.03
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.03
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.03
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.03
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.03
S(ok(z0)) → c30(S(z0)) 219.93/72.03
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.03
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.93/72.03
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.03
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.03
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.03
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.03
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.03
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.03
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.03
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.03
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.03
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.03
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.03
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.03
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.03
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.03
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.03
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.03
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.03
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.03
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.03
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.03
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.03
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.03
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.03
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.03
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.03
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.03
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.03
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.03
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.03
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.03
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.03
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.03
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.03
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.03
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.03
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.03
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.03
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.03
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.03
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.03
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.03
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.03
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.03
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.03
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.03
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.03
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.03
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.03
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.03
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.03
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.03
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.03
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.03
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.03
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.03
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.03
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.03
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.03
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.03
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.03
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.03
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.03
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.03
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.03
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.03
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.03
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.03
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.03
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.03
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.03
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.03
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.03
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.03
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.03
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.03
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.03
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.03
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.03
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.03
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.03
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(length(0)) → c28(LENGTH(ok(0)), PROPER(0)) 219.93/72.06
PROPER(length(true)) → c28(LENGTH(ok(true)), PROPER(true)) 219.93/72.06
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(length(false)) → c28(LENGTH(ok(false)), PROPER(false)) 219.93/72.06
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(length(nil)) → c28(LENGTH(ok(nil)), PROPER(nil)) 219.93/72.06
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28

219.93/72.06

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

Removed 4 trailing tuple parts

219.93/72.06

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.06
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.06
active(eq(z0, z1)) → mark(false) 219.93/72.06
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.06
active(take(0, z0)) → mark(nil) 219.93/72.06
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.06
active(length(nil)) → mark(0) 219.93/72.06
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.06
active(inf(z0)) → inf(active(z0)) 219.93/72.06
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.06
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.06
active(length(z0)) → length(active(z0)) 219.93/72.06
inf(mark(z0)) → mark(inf(z0)) 219.93/72.06
inf(ok(z0)) → ok(inf(z0)) 219.93/72.06
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.06
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.06
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.06
length(mark(z0)) → mark(length(z0)) 219.93/72.06
length(ok(z0)) → ok(length(z0)) 219.93/72.06
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.06
proper(0) → ok(0) 219.93/72.06
proper(true) → ok(true) 219.93/72.06
proper(s(z0)) → s(proper(z0)) 219.93/72.06
proper(false) → ok(false) 219.93/72.06
proper(inf(z0)) → inf(proper(z0)) 219.93/72.06
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.06
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.06
proper(nil) → ok(nil) 219.93/72.06
proper(length(z0)) → length(proper(z0)) 219.93/72.06
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.06
s(ok(z0)) → ok(s(z0)) 219.93/72.06
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.06
top(mark(z0)) → top(proper(z0)) 219.93/72.06
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.06
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.06
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.06
INF(mark(z0)) → c12(INF(z0)) 219.93/72.06
INF(ok(z0)) → c13(INF(z0)) 219.93/72.06
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.06
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.06
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.06
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.06
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.06
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.06
S(ok(z0)) → c30(S(z0)) 219.93/72.06
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.06
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.93/72.06
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.06
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.06
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.06
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.06
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.06
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.06
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.06
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.06
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.06
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.06
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.06
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.06
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.06
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.06
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.06
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.06
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.06
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.06
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.06
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.06
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.06
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.06
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.06
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.06
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.06
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.06
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.06
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.06
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.06
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.06
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.06
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.06
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.06
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.06
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.06
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.06
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.06
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.06
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.06
PROPER(length(nil)) → c28(LENGTH(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.06
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.06
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.06
INF(mark(z0)) → c12(INF(z0)) 219.93/72.06
INF(ok(z0)) → c13(INF(z0)) 219.93/72.06
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.06
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.06
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.06
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.06
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.06
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.06
S(ok(z0)) → c30(S(z0)) 219.93/72.06
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.06
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) 219.93/72.06
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.06
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.06
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.06
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.06
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.06
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.06
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.06
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.06
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.06
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.06
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.06
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.06
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.06
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.06
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.06
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.06
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.06
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.06
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.06
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.06
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.06
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.06
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.06
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.06
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.06
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.06
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.06
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.06
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.06
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.06
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.06
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.06
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.06
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.06
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.06
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.06
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.06
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.06
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.06
PROPER(length(nil)) → c28(LENGTH(ok(nil)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28, c28

219.93/72.06

(43) 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))) 219.93/72.06
TOP(mark(0)) → c32(TOP(ok(0)), PROPER(0)) 219.93/72.06
TOP(mark(true)) → c32(TOP(ok(true)), PROPER(true)) 219.93/72.06
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
TOP(mark(false)) → c32(TOP(ok(false)), PROPER(false)) 219.93/72.06
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
TOP(mark(nil)) → c32(TOP(ok(nil)), PROPER(nil)) 219.93/72.06
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))

219.93/72.06

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.06
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.06
active(eq(z0, z1)) → mark(false) 219.93/72.06
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.06
active(take(0, z0)) → mark(nil) 219.93/72.06
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.06
active(length(nil)) → mark(0) 219.93/72.06
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.06
active(inf(z0)) → inf(active(z0)) 219.93/72.06
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.06
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.06
active(length(z0)) → length(active(z0)) 219.93/72.06
inf(mark(z0)) → mark(inf(z0)) 219.93/72.06
inf(ok(z0)) → ok(inf(z0)) 219.93/72.06
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.06
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.06
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.06
length(mark(z0)) → mark(length(z0)) 219.93/72.06
length(ok(z0)) → ok(length(z0)) 219.93/72.06
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.06
proper(0) → ok(0) 219.93/72.06
proper(true) → ok(true) 219.93/72.06
proper(s(z0)) → s(proper(z0)) 219.93/72.06
proper(false) → ok(false) 219.93/72.06
proper(inf(z0)) → inf(proper(z0)) 219.93/72.06
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.06
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.06
proper(nil) → ok(nil) 219.93/72.06
proper(length(z0)) → length(proper(z0)) 219.93/72.06
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.06
s(ok(z0)) → ok(s(z0)) 219.93/72.06
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.06
top(mark(z0)) → top(proper(z0)) 219.93/72.06
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.06
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.06
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.06
INF(mark(z0)) → c12(INF(z0)) 219.93/72.06
INF(ok(z0)) → c13(INF(z0)) 219.93/72.06
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.06
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.06
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.06
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.06
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.06
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.06
S(ok(z0)) → c30(S(z0)) 219.93/72.06
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.06
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.06
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.06
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.06
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.06
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.06
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.06
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.06
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.06
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.06
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.06
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.06
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.06
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.06
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.06
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.06
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.06
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.06
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.06
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.06
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.06
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.06
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.06
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.06
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.06
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.06
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.06
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.06
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.06
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.06
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.06
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.06
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.06
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.06
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.06
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.06
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.06
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.06
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.06
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.06
PROPER(length(nil)) → c28(LENGTH(ok(nil))) 219.93/72.06
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
TOP(mark(0)) → c32(TOP(ok(0)), PROPER(0)) 219.93/72.06
TOP(mark(true)) → c32(TOP(ok(true)), PROPER(true)) 219.93/72.06
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
TOP(mark(false)) → c32(TOP(ok(false)), PROPER(false)) 219.93/72.06
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
TOP(mark(nil)) → c32(TOP(ok(nil)), PROPER(nil)) 219.93/72.06
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.06
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.06
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.06
INF(mark(z0)) → c12(INF(z0)) 219.93/72.06
INF(ok(z0)) → c13(INF(z0)) 219.93/72.06
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.06
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.06
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.06
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.06
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.06
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.06
S(ok(z0)) → c30(S(z0)) 219.93/72.06
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.06
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.06
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.06
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.06
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.06
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.06
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.06
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.06
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.06
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.06
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.06
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.06
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.06
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.06
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.06
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.06
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.06
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.06
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.06
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.06
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.06
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.06
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.06
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.06
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.06
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.06
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.06
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.06
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.06
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.06
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.06
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.06
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.06
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.06
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.06
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.06
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.06
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.06
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.06
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.06
PROPER(length(nil)) → c28(LENGTH(ok(nil))) 219.93/72.06
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
TOP(mark(0)) → c32(TOP(ok(0)), PROPER(0)) 219.93/72.06
TOP(mark(true)) → c32(TOP(ok(true)), PROPER(true)) 219.93/72.06
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
TOP(mark(false)) → c32(TOP(ok(false)), PROPER(false)) 219.93/72.06
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
TOP(mark(nil)) → c32(TOP(ok(nil)), PROPER(nil)) 219.93/72.06
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28, c28, c32

219.93/72.06

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

Removed 4 trailing tuple parts

219.93/72.06

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.06
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.06
active(eq(z0, z1)) → mark(false) 219.93/72.06
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.06
active(take(0, z0)) → mark(nil) 219.93/72.06
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.06
active(length(nil)) → mark(0) 219.93/72.06
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.06
active(inf(z0)) → inf(active(z0)) 219.93/72.06
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.06
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.06
active(length(z0)) → length(active(z0)) 219.93/72.06
inf(mark(z0)) → mark(inf(z0)) 219.93/72.06
inf(ok(z0)) → ok(inf(z0)) 219.93/72.06
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.06
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.06
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.06
length(mark(z0)) → mark(length(z0)) 219.93/72.06
length(ok(z0)) → ok(length(z0)) 219.93/72.06
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.06
proper(0) → ok(0) 219.93/72.06
proper(true) → ok(true) 219.93/72.06
proper(s(z0)) → s(proper(z0)) 219.93/72.06
proper(false) → ok(false) 219.93/72.06
proper(inf(z0)) → inf(proper(z0)) 219.93/72.06
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.06
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.06
proper(nil) → ok(nil) 219.93/72.06
proper(length(z0)) → length(proper(z0)) 219.93/72.06
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.06
s(ok(z0)) → ok(s(z0)) 219.93/72.06
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.06
top(mark(z0)) → top(proper(z0)) 219.93/72.06
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.06
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.06
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.06
INF(mark(z0)) → c12(INF(z0)) 219.93/72.06
INF(ok(z0)) → c13(INF(z0)) 219.93/72.06
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.06
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.06
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.06
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.06
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.06
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.06
S(ok(z0)) → c30(S(z0)) 219.93/72.06
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.06
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.06
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.06
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.06
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.06
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.06
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.06
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.06
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.06
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.06
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.06
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.06
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.06
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.06
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.06
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.06
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.06
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.06
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.06
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.06
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.06
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.06
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.06
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.06
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.06
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.06
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.06
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.06
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.06
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.06
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.06
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.06
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.06
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.06
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.06
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.06
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.06
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.06
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.06
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.06
PROPER(length(nil)) → c28(LENGTH(ok(nil))) 219.93/72.06
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
TOP(mark(0)) → c32(TOP(ok(0))) 219.93/72.06
TOP(mark(true)) → c32(TOP(ok(true))) 219.93/72.06
TOP(mark(false)) → c32(TOP(ok(false))) 219.93/72.06
TOP(mark(nil)) → c32(TOP(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.06
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.06
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.06
INF(mark(z0)) → c12(INF(z0)) 219.93/72.06
INF(ok(z0)) → c13(INF(z0)) 219.93/72.06
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.06
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.06
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.06
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.06
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.06
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.06
S(ok(z0)) → c30(S(z0)) 219.93/72.06
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.06
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.06
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.06
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.06
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.06
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.06
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.06
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.06
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.06
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.06
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.06
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.06
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.06
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.06
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.06
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.06
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.06
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.06
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.06
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.06
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.06
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.06
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.06
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.06
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.06
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.06
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.06
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.06
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.06
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.06
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.06
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.06
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.06
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.06
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.06
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.06
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.06
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.06
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.06
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.06
PROPER(length(nil)) → c28(LENGTH(ok(nil))) 219.93/72.06
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
TOP(mark(0)) → c32(TOP(ok(0))) 219.93/72.06
TOP(mark(true)) → c32(TOP(ok(true))) 219.93/72.06
TOP(mark(false)) → c32(TOP(ok(false))) 219.93/72.06
TOP(mark(nil)) → c32(TOP(ok(nil)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28, c28, c32, c32

219.93/72.06

(47) 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))) 219.93/72.06
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
TOP(mark(nil)) → c32(TOP(ok(nil)))

We considered the (Usable) Rules:

proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.06
proper(0) → ok(0) 219.93/72.06
proper(true) → ok(true) 219.93/72.06
proper(s(z0)) → s(proper(z0)) 219.93/72.06
proper(false) → ok(false) 219.93/72.06
proper(inf(z0)) → inf(proper(z0)) 219.93/72.06
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.06
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.06
proper(nil) → ok(nil) 219.93/72.06
proper(length(z0)) → length(proper(z0)) 219.93/72.06
length(mark(z0)) → mark(length(z0)) 219.93/72.06
length(ok(z0)) → ok(length(z0)) 219.93/72.06
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.06
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.06
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.06
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.06
inf(mark(z0)) → mark(inf(z0)) 219.93/72.06
inf(ok(z0)) → ok(inf(z0)) 219.93/72.06
s(ok(z0)) → ok(s(z0)) 219.93/72.06
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.06
active(eq(0, 0)) → mark(true) 219.93/72.06
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.06
active(eq(z0, z1)) → mark(false) 219.93/72.06
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.06
active(take(0, z0)) → mark(nil) 219.93/72.06
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.06
active(length(nil)) → mark(0) 219.93/72.06
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.06
active(inf(z0)) → inf(active(z0)) 219.93/72.06
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.06
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.06
active(length(z0)) → length(active(z0))

And the Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.06
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.06
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.06
INF(mark(z0)) → c12(INF(z0)) 219.93/72.06
INF(ok(z0)) → c13(INF(z0)) 219.93/72.06
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.06
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.06
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.06
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.06
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.06
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.06
S(ok(z0)) → c30(S(z0)) 219.93/72.06
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.06
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.06
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.06
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.06
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.06
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.06
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.06
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.06
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.06
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.06
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.06
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.06
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.06
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.06
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.06
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.06
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.06
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.06
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.06
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.06
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.06
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.06
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.06
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.06
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.06
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.06
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.06
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.06
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.06
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.06
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.06
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.06
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.06
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.06
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.06
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.06
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.06
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.06
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.06
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.06
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.06
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.06
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.06
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.06
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.06
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.06
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.06
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.06
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.06
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.06
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.06
PROPER(length(nil)) → c28(LENGTH(ok(nil))) 219.93/72.06
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.06
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 219.93/72.06
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.06
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.06
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.06
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 219.93/72.06
TOP(mark(0)) → c32(TOP(ok(0))) 219.93/72.06
TOP(mark(true)) → c32(TOP(ok(true))) 219.93/72.06
TOP(mark(false)) → c32(TOP(ok(false))) 219.93/72.06
TOP(mark(nil)) → c32(TOP(ok(nil)))

The order we found is given by the following interpretation:
Polynomial interpretation : 219.93/72.06

POL(0) = [1] 219.93/72.06
POL(ACTIVE(x₁)) = 0 219.93/72.06
POL(CONS(x₁, x₂)) = 0 219.93/72.06
POL(EQ(x₁, x₂)) = 0 219.93/72.06
POL(INF(x₁)) = 0 219.93/72.06
POL(LENGTH(x₁)) = 0 219.93/72.06
POL(PROPER(x₁)) = 0 219.93/72.06
POL(S(x₁)) = 0 219.93/72.06
POL(TAKE(x₁, x₂)) = 0 219.93/72.06
POL(TOP(x₁)) = [2]x₁ 219.93/72.06
POL(active(x₁)) = x₁ 219.93/72.06
POL(c(x₁)) = x₁ 219.93/72.06
POL(c1(x₁)) = x₁ 219.93/72.06
POL(c10(x₁, x₂)) = x₁ + x₂ 219.93/72.06
POL(c11(x₁)) = x₁ 219.93/72.06
POL(c11(x₁, x₂)) = x₁ + x₂ 219.93/72.06
POL(c12(x₁)) = x₁ 219.93/72.06
POL(c13(x₁)) = x₁ 219.93/72.06
POL(c14(x₁)) = x₁ 219.93/72.06
POL(c15(x₁)) = x₁ 219.93/72.06
POL(c16(x₁)) = x₁ 219.93/72.06
POL(c17(x₁)) = x₁ 219.93/72.06
POL(c18(x₁)) = x₁ 219.93/72.06
POL(c19(x₁, x₂)) = x₁ + x₂ 219.93/72.06
POL(c19(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 219.93/72.06
POL(c2(x₁)) = x₁ 219.93/72.07
POL(c22(x₁)) = x₁ 219.93/72.07
POL(c22(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c24(x₁)) = x₁ 219.93/72.07
POL(c24(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c25(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c25(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 219.93/72.07
POL(c26(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c26(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 219.93/72.07
POL(c28(x₁)) = x₁ 219.93/72.07
POL(c28(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c29(x₁)) = x₁ 219.93/72.07
POL(c3(x₁)) = x₁ 219.93/72.07
POL(c30(x₁)) = x₁ 219.93/72.07
POL(c31(x₁)) = x₁ 219.93/72.07
POL(c32(x₁)) = x₁ 219.93/72.07
POL(c32(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c33(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c4(x₁)) = x₁ 219.93/72.07
POL(c8(x₁)) = x₁ 219.93/72.07
POL(c8(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c9(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(cons(x₁, x₂)) = 0 219.93/72.07
POL(eq(x₁, x₂)) = [1] 219.93/72.07
POL(false) = [1] 219.93/72.07
POL(inf(x₁)) = [1] 219.93/72.07
POL(length(x₁)) = [1] 219.93/72.07
POL(mark(x₁)) = [1] 219.93/72.07
POL(nil) = 0 219.93/72.07
POL(ok(x₁)) = x₁ 219.93/72.07
POL(proper(x₁)) = 0 219.93/72.07
POL(s(x₁)) = 0 219.93/72.07
POL(take(x₁, x₂)) = [1] 219.93/72.07
POL(true) = [1]

219.93/72.07

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.07
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.07
active(eq(z0, z1)) → mark(false) 219.93/72.07
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.07
active(take(0, z0)) → mark(nil) 219.93/72.07
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.07
active(length(nil)) → mark(0) 219.93/72.07
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.07
active(inf(z0)) → inf(active(z0)) 219.93/72.07
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.07
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.07
active(length(z0)) → length(active(z0)) 219.93/72.07
inf(mark(z0)) → mark(inf(z0)) 219.93/72.07
inf(ok(z0)) → ok(inf(z0)) 219.93/72.07
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.07
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.07
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.07
length(mark(z0)) → mark(length(z0)) 219.93/72.07
length(ok(z0)) → ok(length(z0)) 219.93/72.07
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.07
proper(0) → ok(0) 219.93/72.07
proper(true) → ok(true) 219.93/72.07
proper(s(z0)) → s(proper(z0)) 219.93/72.07
proper(false) → ok(false) 219.93/72.07
proper(inf(z0)) → inf(proper(z0)) 219.93/72.07
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.07
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.07
proper(nil) → ok(nil) 219.93/72.07
proper(length(z0)) → length(proper(z0)) 219.93/72.07
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.07
s(ok(z0)) → ok(s(z0)) 219.93/72.07
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.07
top(mark(z0)) → top(proper(z0)) 219.93/72.07
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.07
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.07
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.07
INF(mark(z0)) → c12(INF(z0)) 219.93/72.07
INF(ok(z0)) → c13(INF(z0)) 219.93/72.07
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.07
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.07
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.07
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.07
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.07
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.07
S(ok(z0)) → c30(S(z0)) 219.93/72.07
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.07
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.07
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.07
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.07
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.07
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.07
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.07
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.07
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.07
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.07
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.07
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.07
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.07
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.07
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.07
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.07
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.07
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.07
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.07
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.07
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.07
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.07
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.07
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.07
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.07
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.07
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.07
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.07
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.07
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.07
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.07
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.07
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.07
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.07
PROPER(length(nil)) → c28(LENGTH(ok(nil))) 219.93/72.07
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
TOP(mark(0)) → c32(TOP(ok(0))) 219.93/72.07
TOP(mark(true)) → c32(TOP(ok(true))) 219.93/72.07
TOP(mark(false)) → c32(TOP(ok(false))) 219.93/72.07
TOP(mark(nil)) → c32(TOP(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.07
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.07
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.07
INF(mark(z0)) → c12(INF(z0)) 219.93/72.07
INF(ok(z0)) → c13(INF(z0)) 219.93/72.07
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.07
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.07
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.07
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.07
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.07
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.07
S(ok(z0)) → c30(S(z0)) 219.93/72.07
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.07
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.07
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.07
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.07
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.07
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.07
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.07
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.07
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.07
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.07
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.07
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.07
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.07
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.07
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.07
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.07
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.07
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.07
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.07
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.07
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.07
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.07
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.07
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.07
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.07
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.07
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.07
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.07
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.07
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.07
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.07
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.07
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.07
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.07
PROPER(length(nil)) → c28(LENGTH(ok(nil))) 219.93/72.07
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
TOP(mark(0)) → c32(TOP(ok(0))) 219.93/72.07
TOP(mark(true)) → c32(TOP(ok(true))) 219.93/72.07
TOP(mark(false)) → c32(TOP(ok(false)))

K tuples:

TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
TOP(mark(nil)) → c32(TOP(ok(nil)))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28, c28, c32, c32

219.93/72.07

(49) 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(0)) → c32(TOP(ok(0))) 219.93/72.07
TOP(mark(true)) → c32(TOP(ok(true))) 219.93/72.07
TOP(mark(false)) → c32(TOP(ok(false)))

We considered the (Usable) Rules:

proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.07
proper(0) → ok(0) 219.93/72.07
proper(true) → ok(true) 219.93/72.07
proper(s(z0)) → s(proper(z0)) 219.93/72.07
proper(false) → ok(false) 219.93/72.07
proper(inf(z0)) → inf(proper(z0)) 219.93/72.07
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.07
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.07
proper(nil) → ok(nil) 219.93/72.07
proper(length(z0)) → length(proper(z0)) 219.93/72.07
length(mark(z0)) → mark(length(z0)) 219.93/72.07
length(ok(z0)) → ok(length(z0)) 219.93/72.07
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.07
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.07
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.07
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.07
inf(mark(z0)) → mark(inf(z0)) 219.93/72.07
inf(ok(z0)) → ok(inf(z0)) 219.93/72.07
s(ok(z0)) → ok(s(z0)) 219.93/72.07
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.07
active(eq(0, 0)) → mark(true) 219.93/72.07
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.07
active(eq(z0, z1)) → mark(false) 219.93/72.07
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.07
active(take(0, z0)) → mark(nil) 219.93/72.07
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.07
active(length(nil)) → mark(0) 219.93/72.07
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.07
active(inf(z0)) → inf(active(z0)) 219.93/72.07
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.07
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.07
active(length(z0)) → length(active(z0))

And the Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.07
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.07
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.07
INF(mark(z0)) → c12(INF(z0)) 219.93/72.07
INF(ok(z0)) → c13(INF(z0)) 219.93/72.07
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.07
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.07
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.07
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.07
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.07
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.07
S(ok(z0)) → c30(S(z0)) 219.93/72.07
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.07
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.07
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.07
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.07
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.07
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.07
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.07
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.07
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.07
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.07
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.07
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.07
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.07
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.07
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.07
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.07
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.07
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.07
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.07
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.07
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.07
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.07
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.07
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.07
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.07
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.07
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.07
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.07
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.07
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.07
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.07
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.07
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.07
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.07
PROPER(length(nil)) → c28(LENGTH(ok(nil))) 219.93/72.07
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
TOP(mark(0)) → c32(TOP(ok(0))) 219.93/72.07
TOP(mark(true)) → c32(TOP(ok(true))) 219.93/72.07
TOP(mark(false)) → c32(TOP(ok(false))) 219.93/72.07
TOP(mark(nil)) → c32(TOP(ok(nil)))

The order we found is given by the following interpretation:
Polynomial interpretation : 219.93/72.07

POL(0) = 0 219.93/72.07
POL(ACTIVE(x₁)) = 0 219.93/72.07
POL(CONS(x₁, x₂)) = 0 219.93/72.07
POL(EQ(x₁, x₂)) = 0 219.93/72.07
POL(INF(x₁)) = 0 219.93/72.07
POL(LENGTH(x₁)) = 0 219.93/72.07
POL(PROPER(x₁)) = 0 219.93/72.07
POL(S(x₁)) = 0 219.93/72.07
POL(TAKE(x₁, x₂)) = 0 219.93/72.07
POL(TOP(x₁)) = x₁ 219.93/72.07
POL(active(x₁)) = x₁ 219.93/72.07
POL(c(x₁)) = x₁ 219.93/72.07
POL(c1(x₁)) = x₁ 219.93/72.07
POL(c10(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c11(x₁)) = x₁ 219.93/72.07
POL(c11(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c12(x₁)) = x₁ 219.93/72.07
POL(c13(x₁)) = x₁ 219.93/72.07
POL(c14(x₁)) = x₁ 219.93/72.07
POL(c15(x₁)) = x₁ 219.93/72.07
POL(c16(x₁)) = x₁ 219.93/72.07
POL(c17(x₁)) = x₁ 219.93/72.07
POL(c18(x₁)) = x₁ 219.93/72.07
POL(c19(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c19(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 219.93/72.07
POL(c2(x₁)) = x₁ 219.93/72.07
POL(c22(x₁)) = x₁ 219.93/72.07
POL(c22(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c24(x₁)) = x₁ 219.93/72.07
POL(c24(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c25(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c25(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 219.93/72.07
POL(c26(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c26(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 219.93/72.07
POL(c28(x₁)) = x₁ 219.93/72.07
POL(c28(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c29(x₁)) = x₁ 219.93/72.07
POL(c3(x₁)) = x₁ 219.93/72.07
POL(c30(x₁)) = x₁ 219.93/72.07
POL(c31(x₁)) = x₁ 219.93/72.07
POL(c32(x₁)) = x₁ 219.93/72.07
POL(c32(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c33(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c4(x₁)) = x₁ 219.93/72.07
POL(c8(x₁)) = x₁ 219.93/72.07
POL(c8(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(c9(x₁, x₂)) = x₁ + x₂ 219.93/72.07
POL(cons(x₁, x₂)) = 0 219.93/72.07
POL(eq(x₁, x₂)) = [2] 219.93/72.07
POL(false) = 0 219.93/72.07
POL(inf(x₁)) = [2] 219.93/72.07
POL(length(x₁)) = [2] 219.93/72.07
POL(mark(x₁)) = [2] 219.93/72.07
POL(nil) = 0 219.93/72.07
POL(ok(x₁)) = x₁ 219.93/72.07
POL(proper(x₁)) = 0 219.93/72.07
POL(s(x₁)) = 0 219.93/72.07
POL(take(x₁, x₂)) = [2] 219.93/72.07
POL(true) = 0

219.93/72.07

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.07
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.07
active(eq(z0, z1)) → mark(false) 219.93/72.07
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.07
active(take(0, z0)) → mark(nil) 219.93/72.07
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.07
active(length(nil)) → mark(0) 219.93/72.07
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.07
active(inf(z0)) → inf(active(z0)) 219.93/72.07
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.07
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.07
active(length(z0)) → length(active(z0)) 219.93/72.07
inf(mark(z0)) → mark(inf(z0)) 219.93/72.07
inf(ok(z0)) → ok(inf(z0)) 219.93/72.07
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.07
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.07
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.07
length(mark(z0)) → mark(length(z0)) 219.93/72.07
length(ok(z0)) → ok(length(z0)) 219.93/72.07
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.07
proper(0) → ok(0) 219.93/72.07
proper(true) → ok(true) 219.93/72.07
proper(s(z0)) → s(proper(z0)) 219.93/72.07
proper(false) → ok(false) 219.93/72.07
proper(inf(z0)) → inf(proper(z0)) 219.93/72.07
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.07
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.07
proper(nil) → ok(nil) 219.93/72.07
proper(length(z0)) → length(proper(z0)) 219.93/72.07
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.07
s(ok(z0)) → ok(s(z0)) 219.93/72.07
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.07
top(mark(z0)) → top(proper(z0)) 219.93/72.07
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.07
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.07
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.07
INF(mark(z0)) → c12(INF(z0)) 219.93/72.07
INF(ok(z0)) → c13(INF(z0)) 219.93/72.07
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.07
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.07
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.07
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.07
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.07
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.07
S(ok(z0)) → c30(S(z0)) 219.93/72.07
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.07
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.07
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.07
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.07
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.07
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.07
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.07
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.07
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.07
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.07
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.07
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.07
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.07
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.07
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.07
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.07
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.07
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.07
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.07
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.07
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.07
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.07
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.07
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.07
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.07
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.07
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.07
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.07
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.07
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.07
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.07
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.07
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.07
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.07
PROPER(length(nil)) → c28(LENGTH(ok(nil))) 219.93/72.07
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
TOP(mark(0)) → c32(TOP(ok(0))) 219.93/72.07
TOP(mark(true)) → c32(TOP(ok(true))) 219.93/72.07
TOP(mark(false)) → c32(TOP(ok(false))) 219.93/72.07
TOP(mark(nil)) → c32(TOP(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.07
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.07
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.07
INF(mark(z0)) → c12(INF(z0)) 219.93/72.07
INF(ok(z0)) → c13(INF(z0)) 219.93/72.07
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.07
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.07
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.07
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.07
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.07
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.07
S(ok(z0)) → c30(S(z0)) 219.93/72.07
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.07
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) 219.93/72.07
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.07
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.07
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.07
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.07
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.07
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.07
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.07
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.07
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.07
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.07
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.07
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.07
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.07
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.07
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.07
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.07
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.07
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.07
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.07
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.07
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.07
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.07
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.07
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.07
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.07
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.07
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.07
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.07
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.07
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.07
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.07
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.07
PROPER(length(nil)) → c28(LENGTH(ok(nil))) 219.93/72.07
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))

K tuples:

TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
TOP(mark(nil)) → c32(TOP(ok(nil))) 219.93/72.07
TOP(mark(0)) → c32(TOP(ok(0))) 219.93/72.07
TOP(mark(true)) → c32(TOP(ok(true))) 219.93/72.07
TOP(mark(false)) → c32(TOP(ok(false)))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28, c28, c32, c32

219.93/72.07

(51) 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))) 219.93/72.07
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1))) 219.93/72.07
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil))) 219.93/72.07
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.93/72.07
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))

219.93/72.07

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.07
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.07
active(eq(z0, z1)) → mark(false) 219.93/72.07
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.07
active(take(0, z0)) → mark(nil) 219.93/72.07
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.07
active(length(nil)) → mark(0) 219.93/72.07
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.07
active(inf(z0)) → inf(active(z0)) 219.93/72.07
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.07
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.07
active(length(z0)) → length(active(z0)) 219.93/72.07
inf(mark(z0)) → mark(inf(z0)) 219.93/72.07
inf(ok(z0)) → ok(inf(z0)) 219.93/72.07
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.07
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.07
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.07
length(mark(z0)) → mark(length(z0)) 219.93/72.07
length(ok(z0)) → ok(length(z0)) 219.93/72.07
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.07
proper(0) → ok(0) 219.93/72.07
proper(true) → ok(true) 219.93/72.07
proper(s(z0)) → s(proper(z0)) 219.93/72.07
proper(false) → ok(false) 219.93/72.07
proper(inf(z0)) → inf(proper(z0)) 219.93/72.07
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.07
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.07
proper(nil) → ok(nil) 219.93/72.07
proper(length(z0)) → length(proper(z0)) 219.93/72.07
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.07
s(ok(z0)) → ok(s(z0)) 219.93/72.07
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.07
top(mark(z0)) → top(proper(z0)) 219.93/72.07
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.07
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.07
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.07
INF(mark(z0)) → c12(INF(z0)) 219.93/72.07
INF(ok(z0)) → c13(INF(z0)) 219.93/72.07
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.07
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.07
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.07
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.07
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.07
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.07
S(ok(z0)) → c30(S(z0)) 219.93/72.07
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.07
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.07
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.07
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.07
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.07
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.07
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.07
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.07
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.07
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.07
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.07
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.07
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.07
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.07
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.07
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.07
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.07
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.07
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.07
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.07
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.07
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.07
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.07
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.07
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.07
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.07
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.07
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.07
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.07
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.07
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.07
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.07
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.07
PROPER(length(nil)) → c28(LENGTH(ok(nil))) 219.93/72.07
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
TOP(mark(0)) → c32(TOP(ok(0))) 219.93/72.07
TOP(mark(true)) → c32(TOP(ok(true))) 219.93/72.07
TOP(mark(false)) → c32(TOP(ok(false))) 219.93/72.07
TOP(mark(nil)) → c32(TOP(ok(nil))) 219.93/72.07
TOP(ok(eq(0, 0))) → c33(TOP(mark(true)), ACTIVE(eq(0, 0))) 219.93/72.07
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1))) 219.93/72.07
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil))) 219.93/72.07
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.93/72.07
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1)) 219.93/72.07
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.07
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.07
INF(mark(z0)) → c12(INF(z0)) 219.93/72.07
INF(ok(z0)) → c13(INF(z0)) 219.93/72.07
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.07
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.07
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.07
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.07
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.07
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.07
S(ok(z0)) → c30(S(z0)) 219.93/72.07
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.07
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.07
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.07
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.07
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.07
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.07
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.07
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.07
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.07
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.07
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.07
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.07
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.07
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.07
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.07
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.07
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.07
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.07
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.07
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.07
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.07
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.07
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.07
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.07
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.07
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.07
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.07
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.07
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.07
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.07
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.07
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.07
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.07
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.07
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.07
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.07
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.07
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.07
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.07
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.07
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.07
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.07
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.07
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.07
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.07
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.07
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.07
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.07
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.07
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.07
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.08
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.08
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.08
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.08
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.08
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.08
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.08
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.08
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.08
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.08
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.08
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.08
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.08
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.08
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.08
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.08
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.08
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.08
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.08
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.08
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.08
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.08
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.08
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.08
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.08
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.08
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.08
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.08
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.08
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.08
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.08
PROPER(length(nil)) → c28(LENGTH(ok(nil))) 219.93/72.08
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.08
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.08
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.08
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 219.93/72.08
TOP(ok(eq(0, 0))) → c33(TOP(mark(true)), ACTIVE(eq(0, 0))) 219.93/72.08
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 219.93/72.08
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1))) 219.93/72.08
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.08
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.08
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.08
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil))) 219.93/72.08
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.93/72.08
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.08
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.08
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.08
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))

K tuples:

TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 219.93/72.08
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.08
TOP(mark(nil)) → c32(TOP(ok(nil))) 219.93/72.08
TOP(mark(0)) → c32(TOP(ok(0))) 219.93/72.08
TOP(mark(true)) → c32(TOP(ok(true))) 219.93/72.08
TOP(mark(false)) → c32(TOP(ok(false)))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28, c28, c32, c32, c33

219.93/72.08

(53) 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)) 219.93/72.08
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0)) 219.93/72.08
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1)) 219.93/72.08
ACTIVE(inf(z0)) → c3(S(z0)) 219.93/72.08
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2))) 219.93/72.08
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2)) 219.93/72.08
ACTIVE(length(cons(z0, z1))) → c(S(length(z1))) 219.93/72.08
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1)) 219.93/72.08
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.08
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.08
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.08
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.08
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.08
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.08
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0))) 219.93/72.08
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true))) 219.93/72.08
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1)))) 219.93/72.08
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.08
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false))) 219.93/72.08
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1))) 219.93/72.08
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.08
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.08
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.08
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.08
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.08
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.08
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0))) 219.93/72.08
ACTIVE(inf(length(nil))) → c8(INF(mark(0))) 219.93/72.08
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true))) 219.93/72.08
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0))) 219.93/72.08
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1))))) 219.93/72.08
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.08
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1)))) 219.93/72.08
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1)))) 219.93/72.08
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false))) 219.93/72.08
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1))) 219.93/72.08
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1))))) 219.93/72.08
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1)))) 219.93/72.08
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.08
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.08
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.08
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.08
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.08
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.08
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.08
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.08
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.08
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0)) 219.93/72.08
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0)) 219.93/72.08
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0)) 219.93/72.08
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.08
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.08
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0))) 219.93/72.08
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.08
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.08
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.08
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0))) 219.93/72.08
PROPER(s(0)) → c22(S(ok(0))) 219.93/72.08
PROPER(s(true)) → c22(S(ok(true))) 219.93/72.08
PROPER(s(false)) → c22(S(ok(false))) 219.93/72.08
PROPER(s(nil)) → c22(S(ok(nil))) 219.93/72.08
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.08
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0))) 219.93/72.08
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.08
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.08
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.08
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0))) 219.93/72.08
PROPER(inf(0)) → c24(INF(ok(0))) 219.93/72.08
PROPER(inf(true)) → c24(INF(ok(true))) 219.93/72.08
PROPER(inf(false)) → c24(INF(ok(false))) 219.93/72.08
PROPER(inf(nil)) → c24(INF(ok(nil))) 219.93/72.08
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.08
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.08
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.08
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.08
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.08
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.08
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.08
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.08
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.08
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 219.93/72.08
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0)) 219.93/72.08
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0)) 219.93/72.08
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.08
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1))) 219.93/72.08
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 219.93/72.08
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0))) 219.93/72.08
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 219.93/72.08
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1))) 219.93/72.08
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0))) 219.93/72.08
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 219.93/72.08
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1)) 219.93/72.08
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1)) 219.93/72.08
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1)) 219.93/72.08
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0)) 219.93/72.08
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0)) 219.93/72.08
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0)) 219.93/72.08
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0)) 219.93/72.08
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1)) 219.93/72.08
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.08
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0))) 219.93/72.08
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.08
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.08
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.08
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0))) 219.93/72.08
PROPER(length(0)) → c28(LENGTH(ok(0))) 219.93/72.08
PROPER(length(true)) → c28(LENGTH(ok(true))) 219.93/72.08
PROPER(length(false)) → c28(LENGTH(ok(false))) 219.93/72.08
PROPER(length(nil)) → c28(LENGTH(ok(nil))) 219.93/72.08
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1))) 219.93/72.08
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0))) 219.93/72.08
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0))) 219.93/72.08
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 219.93/72.08
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1))) 219.93/72.08
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0))) 219.93/72.08
TOP(ok(eq(0, 0))) → c33(TOP(mark(true)), ACTIVE(eq(0, 0))) 219.93/72.08
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1)))) 219.93/72.08
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1))) 219.93/72.08
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0))) 219.93/72.08
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0))) 219.93/72.08
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2)))) 219.93/72.08
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil))) 219.93/72.08
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1)))) 219.93/72.08
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0))) 219.93/72.08
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1))) 219.93/72.08
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1))) 219.93/72.08
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))

219.93/72.08

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.08
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.08
active(eq(z0, z1)) → mark(false) 219.93/72.08
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.08
active(take(0, z0)) → mark(nil) 219.93/72.08
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.08
active(length(nil)) → mark(0) 219.93/72.08
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.08
active(inf(z0)) → inf(active(z0)) 219.93/72.08
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.08
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.08
active(length(z0)) → length(active(z0)) 219.93/72.08
inf(mark(z0)) → mark(inf(z0)) 219.93/72.08
inf(ok(z0)) → ok(inf(z0)) 219.93/72.08
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.08
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.08
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.08
length(mark(z0)) → mark(length(z0)) 219.93/72.08
length(ok(z0)) → ok(length(z0)) 219.93/72.08
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.08
proper(0) → ok(0) 219.93/72.08
proper(true) → ok(true) 219.93/72.08
proper(s(z0)) → s(proper(z0)) 219.93/72.08
proper(false) → ok(false) 219.93/72.08
proper(inf(z0)) → inf(proper(z0)) 219.93/72.08
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.08
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.08
proper(nil) → ok(nil) 219.93/72.08
proper(length(z0)) → length(proper(z0)) 219.93/72.08
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.08
s(ok(z0)) → ok(s(z0)) 219.93/72.08
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.08
top(mark(z0)) → top(proper(z0)) 219.93/72.08
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.08
TOP(mark(0)) → c32(TOP(ok(0))) 219.93/72.08
TOP(mark(true)) → c32(TOP(ok(true))) 219.93/72.08
TOP(mark(false)) → c32(TOP(ok(false))) 219.93/72.08
TOP(mark(nil)) → c32(TOP(ok(nil)))

S tuples:

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

K tuples:

TOP(mark(nil)) → c32(TOP(ok(nil))) 219.93/72.08
TOP(mark(0)) → c32(TOP(ok(0))) 219.93/72.08
TOP(mark(true)) → c32(TOP(ok(true))) 219.93/72.08
TOP(mark(false)) → c32(TOP(ok(false)))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

219.93/72.08

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

Removed 4 trailing tuple parts

219.93/72.08

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.08
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.08
active(eq(z0, z1)) → mark(false) 219.93/72.08
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.08
active(take(0, z0)) → mark(nil) 219.93/72.08
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.08
active(length(nil)) → mark(0) 219.93/72.08
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.08
active(inf(z0)) → inf(active(z0)) 219.93/72.08
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.08
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.08
active(length(z0)) → length(active(z0)) 219.93/72.08
inf(mark(z0)) → mark(inf(z0)) 219.93/72.08
inf(ok(z0)) → ok(inf(z0)) 219.93/72.08
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.08
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.08
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.08
length(mark(z0)) → mark(length(z0)) 219.93/72.08
length(ok(z0)) → ok(length(z0)) 219.93/72.08
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.08
proper(0) → ok(0) 219.93/72.08
proper(true) → ok(true) 219.93/72.08
proper(s(z0)) → s(proper(z0)) 219.93/72.08
proper(false) → ok(false) 219.93/72.08
proper(inf(z0)) → inf(proper(z0)) 219.93/72.08
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.08
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.08
proper(nil) → ok(nil) 219.93/72.08
proper(length(z0)) → length(proper(z0)) 219.93/72.08
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.08
s(ok(z0)) → ok(s(z0)) 219.93/72.08
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.08
top(mark(z0)) → top(proper(z0)) 219.93/72.08
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1)) 219.93/72.08
TOP(mark(0)) → c32 219.93/72.08
TOP(mark(true)) → c32 219.93/72.08
TOP(mark(false)) → c32 219.93/72.08
TOP(mark(nil)) → c32

S tuples:

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

K tuples:

TOP(mark(0)) → c32 219.93/72.08
TOP(mark(true)) → c32 219.93/72.08
TOP(mark(false)) → c32 219.93/72.08
TOP(mark(nil)) → c32

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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

219.93/72.08

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

Removed 4 trailing nodes:

TOP(mark(false)) → c32 219.93/72.08
TOP(mark(0)) → c32 219.93/72.08
TOP(mark(true)) → c32 219.93/72.08
TOP(mark(nil)) → c32

219.93/72.08

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.08
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.08
active(eq(z0, z1)) → mark(false) 219.93/72.08
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.08
active(take(0, z0)) → mark(nil) 219.93/72.08
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.08
active(length(nil)) → mark(0) 219.93/72.08
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.08
active(inf(z0)) → inf(active(z0)) 219.93/72.08
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.08
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.08
active(length(z0)) → length(active(z0)) 219.93/72.08
inf(mark(z0)) → mark(inf(z0)) 219.93/72.08
inf(ok(z0)) → ok(inf(z0)) 219.93/72.08
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.08
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.08
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.08
length(mark(z0)) → mark(length(z0)) 219.93/72.08
length(ok(z0)) → ok(length(z0)) 219.93/72.08
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.08
proper(0) → ok(0) 219.93/72.08
proper(true) → ok(true) 219.93/72.08
proper(s(z0)) → s(proper(z0)) 219.93/72.08
proper(false) → ok(false) 219.93/72.08
proper(inf(z0)) → inf(proper(z0)) 219.93/72.08
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.08
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.08
proper(nil) → ok(nil) 219.93/72.08
proper(length(z0)) → length(proper(z0)) 219.93/72.08
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.08
s(ok(z0)) → ok(s(z0)) 219.93/72.08
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.08
top(mark(z0)) → top(proper(z0)) 219.93/72.08
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

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

219.93/72.08

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

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

LENGTH(ok(z0)) → c18(LENGTH(z0))

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

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 219.93/72.08

POL(CONS(x₁, x₂)) = 0 219.93/72.08
POL(EQ(x₁, x₂)) = 0 219.93/72.08
POL(INF(x₁)) = 0 219.93/72.08
POL(LENGTH(x₁)) = x₁ 219.93/72.08
POL(S(x₁)) = 0 219.93/72.08
POL(TAKE(x₁, x₂)) = 0 219.93/72.08
POL(c12(x₁)) = x₁ 219.93/72.08
POL(c13(x₁)) = x₁ 219.93/72.08
POL(c14(x₁)) = x₁ 219.93/72.08
POL(c15(x₁)) = x₁ 219.93/72.08
POL(c16(x₁)) = x₁ 219.93/72.08
POL(c17(x₁)) = x₁ 219.93/72.08
POL(c18(x₁)) = x₁ 219.93/72.08
POL(c29(x₁)) = x₁ 219.93/72.08
POL(c30(x₁)) = x₁ 219.93/72.08
POL(c31(x₁)) = x₁ 219.93/72.08
POL(mark(x₁)) = x₁ 219.93/72.08
POL(ok(x₁)) = [1] + x₁

219.93/72.08

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.08
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.08
active(eq(z0, z1)) → mark(false) 219.93/72.08
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.08
active(take(0, z0)) → mark(nil) 219.93/72.08
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.08
active(length(nil)) → mark(0) 219.93/72.08
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.08
active(inf(z0)) → inf(active(z0)) 219.93/72.08
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.08
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.08
active(length(z0)) → length(active(z0)) 219.93/72.08
inf(mark(z0)) → mark(inf(z0)) 219.93/72.08
inf(ok(z0)) → ok(inf(z0)) 219.93/72.08
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.08
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.08
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.08
length(mark(z0)) → mark(length(z0)) 219.93/72.08
length(ok(z0)) → ok(length(z0)) 219.93/72.08
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.08
proper(0) → ok(0) 219.93/72.08
proper(true) → ok(true) 219.93/72.08
proper(s(z0)) → s(proper(z0)) 219.93/72.08
proper(false) → ok(false) 219.93/72.08
proper(inf(z0)) → inf(proper(z0)) 219.93/72.08
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.08
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.08
proper(nil) → ok(nil) 219.93/72.08
proper(length(z0)) → length(proper(z0)) 219.93/72.08
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.08
s(ok(z0)) → ok(s(z0)) 219.93/72.08
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.08
top(mark(z0)) → top(proper(z0)) 219.93/72.08
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

K tuples:

LENGTH(ok(z0)) → c18(LENGTH(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

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

219.93/72.08

(61) 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)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0))

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

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 219.93/72.08

POL(CONS(x₁, x₂)) = 0 219.93/72.08
POL(EQ(x₁, x₂)) = [2]x₂ 219.93/72.08
POL(INF(x₁)) = 0 219.93/72.08
POL(LENGTH(x₁)) = x₁ 219.93/72.08
POL(S(x₁)) = x₁ 219.93/72.08
POL(TAKE(x₁, x₂)) = x₁ 219.93/72.08
POL(c12(x₁)) = x₁ 219.93/72.08
POL(c13(x₁)) = x₁ 219.93/72.08
POL(c14(x₁)) = x₁ 219.93/72.08
POL(c15(x₁)) = x₁ 219.93/72.08
POL(c16(x₁)) = x₁ 219.93/72.08
POL(c17(x₁)) = x₁ 219.93/72.08
POL(c18(x₁)) = x₁ 219.93/72.08
POL(c29(x₁)) = x₁ 219.93/72.08
POL(c30(x₁)) = x₁ 219.93/72.08
POL(c31(x₁)) = x₁ 219.93/72.08
POL(mark(x₁)) = [1] + x₁ 219.93/72.08
POL(ok(x₁)) = [1] + x₁

219.93/72.08

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.08
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.08
active(eq(z0, z1)) → mark(false) 219.93/72.08
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.08
active(take(0, z0)) → mark(nil) 219.93/72.08
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.08
active(length(nil)) → mark(0) 219.93/72.08
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.08
active(inf(z0)) → inf(active(z0)) 219.93/72.08
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.08
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.08
active(length(z0)) → length(active(z0)) 219.93/72.08
inf(mark(z0)) → mark(inf(z0)) 219.93/72.08
inf(ok(z0)) → ok(inf(z0)) 219.93/72.08
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.08
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.08
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.08
length(mark(z0)) → mark(length(z0)) 219.93/72.08
length(ok(z0)) → ok(length(z0)) 219.93/72.08
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.08
proper(0) → ok(0) 219.93/72.08
proper(true) → ok(true) 219.93/72.08
proper(s(z0)) → s(proper(z0)) 219.93/72.08
proper(false) → ok(false) 219.93/72.08
proper(inf(z0)) → inf(proper(z0)) 219.93/72.08
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.08
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.08
proper(nil) → ok(nil) 219.93/72.08
proper(length(z0)) → length(proper(z0)) 219.93/72.08
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.08
s(ok(z0)) → ok(s(z0)) 219.93/72.08
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.08
top(mark(z0)) → top(proper(z0)) 219.93/72.08
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

K tuples:

LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

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

219.93/72.08

(63) 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)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))

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

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 219.93/72.08

POL(CONS(x₁, x₂)) = x₁ + x₂ 219.93/72.08
POL(EQ(x₁, x₂)) = [5]x₁ + [5]x₂ 219.93/72.08
POL(INF(x₁)) = [2]x₁ 219.93/72.08
POL(LENGTH(x₁)) = [5]x₁ 219.93/72.08
POL(S(x₁)) = [5]x₁ 219.93/72.08
POL(TAKE(x₁, x₂)) = [3]x₁ + x₂ 219.93/72.08
POL(c12(x₁)) = x₁ 219.93/72.08
POL(c13(x₁)) = x₁ 219.93/72.08
POL(c14(x₁)) = x₁ 219.93/72.08
POL(c15(x₁)) = x₁ 219.93/72.08
POL(c16(x₁)) = x₁ 219.93/72.08
POL(c17(x₁)) = x₁ 219.93/72.08
POL(c18(x₁)) = x₁ 219.93/72.08
POL(c29(x₁)) = x₁ 219.93/72.08
POL(c30(x₁)) = x₁ 219.93/72.08
POL(c31(x₁)) = x₁ 219.93/72.08
POL(mark(x₁)) = [1] + x₁ 219.93/72.08
POL(ok(x₁)) = x₁

219.93/72.08

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.08
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.08
active(eq(z0, z1)) → mark(false) 219.93/72.08
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.08
active(take(0, z0)) → mark(nil) 219.93/72.08
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.08
active(length(nil)) → mark(0) 219.93/72.08
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.08
active(inf(z0)) → inf(active(z0)) 219.93/72.08
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.08
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.08
active(length(z0)) → length(active(z0)) 219.93/72.08
inf(mark(z0)) → mark(inf(z0)) 219.93/72.08
inf(ok(z0)) → ok(inf(z0)) 219.93/72.08
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.08
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.08
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.08
length(mark(z0)) → mark(length(z0)) 219.93/72.08
length(ok(z0)) → ok(length(z0)) 219.93/72.08
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.08
proper(0) → ok(0) 219.93/72.08
proper(true) → ok(true) 219.93/72.08
proper(s(z0)) → s(proper(z0)) 219.93/72.08
proper(false) → ok(false) 219.93/72.08
proper(inf(z0)) → inf(proper(z0)) 219.93/72.08
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.08
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.08
proper(nil) → ok(nil) 219.93/72.08
proper(length(z0)) → length(proper(z0)) 219.93/72.08
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.08
s(ok(z0)) → ok(s(z0)) 219.93/72.08
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.08
top(mark(z0)) → top(proper(z0)) 219.93/72.08
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

K tuples:

LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

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

219.93/72.08

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

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

INF(ok(z0)) → c13(INF(z0))

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

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 219.93/72.08

POL(CONS(x₁, x₂)) = 0 219.93/72.08
POL(EQ(x₁, x₂)) = x₁ + x₂ 219.93/72.08
POL(INF(x₁)) = [2]x₁ 219.93/72.08
POL(LENGTH(x₁)) = [3]x₁ + [3]x₁² 219.93/72.08
POL(S(x₁)) = [3]x₁ + [3]x₁² 219.93/72.08
POL(TAKE(x₁, x₂)) = x₁ + x₂ 219.93/72.08
POL(c12(x₁)) = x₁ 219.93/72.08
POL(c13(x₁)) = x₁ 219.93/72.08
POL(c14(x₁)) = x₁ 219.93/72.08
POL(c15(x₁)) = x₁ 219.93/72.08
POL(c16(x₁)) = x₁ 219.93/72.08
POL(c17(x₁)) = x₁ 219.93/72.08
POL(c18(x₁)) = x₁ 219.93/72.08
POL(c29(x₁)) = x₁ 219.93/72.08
POL(c30(x₁)) = x₁ 219.93/72.08
POL(c31(x₁)) = x₁ 219.93/72.08
POL(mark(x₁)) = [3] + x₁ 219.93/72.08
POL(ok(x₁)) = [2] + x₁

219.93/72.08

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.08
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.08
active(eq(z0, z1)) → mark(false) 219.93/72.08
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.08
active(take(0, z0)) → mark(nil) 219.93/72.08
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.08
active(length(nil)) → mark(0) 219.93/72.08
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.08
active(inf(z0)) → inf(active(z0)) 219.93/72.08
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.08
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.08
active(length(z0)) → length(active(z0)) 219.93/72.08
inf(mark(z0)) → mark(inf(z0)) 219.93/72.08
inf(ok(z0)) → ok(inf(z0)) 219.93/72.08
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.08
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.08
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.08
length(mark(z0)) → mark(length(z0)) 219.93/72.08
length(ok(z0)) → ok(length(z0)) 219.93/72.08
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.08
proper(0) → ok(0) 219.93/72.08
proper(true) → ok(true) 219.93/72.08
proper(s(z0)) → s(proper(z0)) 219.93/72.08
proper(false) → ok(false) 219.93/72.08
proper(inf(z0)) → inf(proper(z0)) 219.93/72.08
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.08
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.08
proper(nil) → ok(nil) 219.93/72.08
proper(length(z0)) → length(proper(z0)) 219.93/72.08
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.08
s(ok(z0)) → ok(s(z0)) 219.93/72.08
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.08
top(mark(z0)) → top(proper(z0)) 219.93/72.08
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

K tuples:

LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

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

219.93/72.08

(67) 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:none
And the Tuples:

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 219.93/72.08

POL(CONS(x₁, x₂)) = x₁ 219.93/72.08
POL(EQ(x₁, x₂)) = [5]x₁ + [3]x₂ 219.93/72.08
POL(INF(x₁)) = [3]x₁ 219.93/72.08
POL(LENGTH(x₁)) = [5]x₁ 219.93/72.08
POL(S(x₁)) = [5]x₁ 219.93/72.08
POL(TAKE(x₁, x₂)) = [3]x₁ + [3]x₂ 219.93/72.08
POL(c12(x₁)) = x₁ 219.93/72.08
POL(c13(x₁)) = x₁ 219.93/72.08
POL(c14(x₁)) = x₁ 219.93/72.08
POL(c15(x₁)) = x₁ 219.93/72.08
POL(c16(x₁)) = x₁ 219.93/72.08
POL(c17(x₁)) = x₁ 219.93/72.08
POL(c18(x₁)) = x₁ 219.93/72.08
POL(c29(x₁)) = x₁ 219.93/72.08
POL(c30(x₁)) = x₁ 219.93/72.08
POL(c31(x₁)) = x₁ 219.93/72.08
POL(mark(x₁)) = x₁ 219.93/72.08
POL(ok(x₁)) = [1] + x₁

219.93/72.08

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true) 219.93/72.08
active(eq(s(z0), s(z1))) → mark(eq(z0, z1)) 219.93/72.08
active(eq(z0, z1)) → mark(false) 219.93/72.08
active(inf(z0)) → mark(cons(z0, inf(s(z0)))) 219.93/72.08
active(take(0, z0)) → mark(nil) 219.93/72.08
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2))) 219.93/72.08
active(length(nil)) → mark(0) 219.93/72.08
active(length(cons(z0, z1))) → mark(s(length(z1))) 219.93/72.08
active(inf(z0)) → inf(active(z0)) 219.93/72.08
active(take(z0, z1)) → take(active(z0), z1) 219.93/72.08
active(take(z0, z1)) → take(z0, active(z1)) 219.93/72.08
active(length(z0)) → length(active(z0)) 219.93/72.08
inf(mark(z0)) → mark(inf(z0)) 219.93/72.08
inf(ok(z0)) → ok(inf(z0)) 219.93/72.08
take(mark(z0), z1) → mark(take(z0, z1)) 219.93/72.08
take(z0, mark(z1)) → mark(take(z0, z1)) 219.93/72.08
take(ok(z0), ok(z1)) → ok(take(z0, z1)) 219.93/72.08
length(mark(z0)) → mark(length(z0)) 219.93/72.08
length(ok(z0)) → ok(length(z0)) 219.93/72.08
proper(eq(z0, z1)) → eq(proper(z0), proper(z1)) 219.93/72.08
proper(0) → ok(0) 219.93/72.08
proper(true) → ok(true) 219.93/72.08
proper(s(z0)) → s(proper(z0)) 219.93/72.08
proper(false) → ok(false) 219.93/72.08
proper(inf(z0)) → inf(proper(z0)) 219.93/72.08
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 219.93/72.08
proper(take(z0, z1)) → take(proper(z0), proper(z1)) 219.93/72.08
proper(nil) → ok(nil) 219.93/72.08
proper(length(z0)) → length(proper(z0)) 219.93/72.08
eq(ok(z0), ok(z1)) → ok(eq(z0, z1)) 219.93/72.08
s(ok(z0)) → ok(s(z0)) 219.93/72.08
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 219.93/72.08
top(mark(z0)) → top(proper(z0)) 219.93/72.08
top(ok(z0)) → top(active(z0))

Tuples:

INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:none
K tuples:

LENGTH(ok(z0)) → c18(LENGTH(z0)) 219.93/72.08
TAKE(mark(z0), z1) → c14(TAKE(z0, z1)) 219.93/72.08
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1)) 219.93/72.08
LENGTH(mark(z0)) → c17(LENGTH(z0)) 219.93/72.08
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1)) 219.93/72.08
S(ok(z0)) → c30(S(z0)) 219.93/72.08
INF(mark(z0)) → c12(INF(z0)) 219.93/72.08
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1)) 219.93/72.08
INF(ok(z0)) → c13(INF(z0)) 219.93/72.08
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

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

219.93/72.08

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

The set S is empty

219.93/72.08

(70) BOUNDS(O(1), O(1))

219.93/72.08