67.45/38.83 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml

67.45/38.83

67.45/38.83
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 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳6 CdtProblem

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

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↳8 CdtProblem

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

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↳10 CdtProblem

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

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↳12 CdtProblem

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

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↳14 CdtProblem

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

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↳16 CdtProblem

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

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↳18 CdtProblem

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

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↳20 CdtProblem

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↳21 CdtLeafRemovalProof (ComplexityIfPolyImplication)

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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 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

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↳28 CdtProblem

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

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↳30 CdtProblem

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

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↳32 CdtProblem

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

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↳34 CdtProblem

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

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↳36 CdtProblem

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

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↳38 CdtProblem

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

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↳40 CdtProblem

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

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↳42 CdtProblem

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

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↳44 CdtProblem

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

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↳46 CdtProblem

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↳47 CdtLeafRemovalProof (ComplexityIfPolyImplication)

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↳48 CdtProblem

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↳49 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))

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↳50 CdtProblem

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

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↳52 CdtProblem

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

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↳54 CdtProblem

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↳55 CdtRewritingProof (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^2))))

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↳62 CdtProblem

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

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↳64 BOUNDS(O(1), O(1))

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67.45/38.83 67.45/38.83

67.45/38.83

(0) Obligation:

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

cond(true, x, y) → cond(gr(x, y), p(x), s(y)) 67.45/38.83
gr(0, x) → false 67.45/38.83
gr(s(x), 0) → true 67.45/38.83
gr(s(x), s(y)) → gr(x, y) 67.45/38.83
p(0) → 0 67.45/38.83
p(s(x)) → 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:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.83
gr(0, z0) → false 67.45/38.83
gr(s(z0), 0) → true 67.45/38.83
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.83
p(0) → 0 67.45/38.83
p(s(z0)) → z0

Tuples:

COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1), P(z0)) 67.45/38.83
GR(s(z0), s(z1)) → c3(GR(z0, z1))

S tuples:

COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1), P(z0)) 67.45/38.83
GR(s(z0), s(z1)) → c3(GR(z0, z1))

K tuples:none
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c3

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

Removed 1 trailing tuple parts

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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.83
gr(0, z0) → false 67.45/38.83
gr(s(z0), 0) → true 67.45/38.83
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.83
p(0) → 0 67.45/38.83
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.83
COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.83
COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1))

K tuples:none
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c

67.45/38.87

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

Use narrowing to replace COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1)) by

COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)), GR(0, x1)) 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) 67.45/38.87
COND(true, 0, z0) → c(COND(false, p(0), s(z0)), GR(0, z0)) 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)), GR(s(z0), 0)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))

67.45/38.87

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)), GR(0, x1)) 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) 67.45/38.87
COND(true, 0, z0) → c(COND(false, p(0), s(z0)), GR(0, z0)) 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)), GR(s(z0), 0)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)), GR(0, x1)) 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) 67.45/38.87
COND(true, 0, z0) → c(COND(false, p(0), s(z0)), GR(0, z0)) 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)), GR(s(z0), 0)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))

K tuples:none
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c

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

Removed 4 trailing tuple parts

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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))

K tuples:none
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

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

Removed 1 trailing nodes:

COND(true, 0, z0) → c

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(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))

K tuples:none
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

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

The following tuples could be moved from S to K by knowledge propagation:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))

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(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

67.45/38.87

(13) 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.

COND(true, 0, z0) → c

We considered the (Usable) Rules:

p(s(z0)) → z0 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1)

And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))

The order we found is given by the following interpretation:
Polynomial interpretation : 67.45/38.87

POL(0) = [5] 67.45/38.87
POL(COND(x₁, x₂, x₃)) = [1] + [2]x₃ 67.45/38.87
POL(GR(x₁, x₂)) = 0 67.45/38.87
POL(c) = 0 67.45/38.87
POL(c(x₁)) = x₁ 67.45/38.87
POL(c(x₁, x₂)) = x₁ + x₂ 67.45/38.87
POL(c3(x₁)) = x₁ 67.45/38.87
POL(false) = 0 67.45/38.87
POL(gr(x₁, x₂)) = 0 67.45/38.87
POL(p(x₁)) = [1] 67.45/38.87
POL(s(x₁)) = 0 67.45/38.87
POL(true) = 0

67.45/38.87

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, 0, z0) → c

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

67.45/38.87

(15) 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.

COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))

We considered the (Usable) Rules:

p(s(z0)) → z0 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1)

And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))

The order we found is given by the following interpretation:
Polynomial interpretation : 67.45/38.87

POL(0) = 0 67.45/38.87
POL(COND(x₁, x₂, x₃)) = [4]x₂ 67.45/38.87
POL(GR(x₁, x₂)) = 0 67.45/38.87
POL(c) = 0 67.45/38.87
POL(c(x₁)) = x₁ 67.45/38.87
POL(c(x₁, x₂)) = x₁ + x₂ 67.45/38.87
POL(c3(x₁)) = x₁ 67.45/38.87
POL(false) = 0 67.45/38.87
POL(gr(x₁, x₂)) = 0 67.45/38.87
POL(p(x₁)) = x₁ 67.45/38.87
POL(s(x₁)) = [2] + x₁ 67.45/38.87
POL(true) = 0

67.45/38.87

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

67.45/38.87

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

Use narrowing to replace COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) by

COND(true, s(z0), 0) → c(COND(true, z0, s(0)), GR(s(z0), 0)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))

67.45/38.87

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, z0, s(0)), GR(s(z0), 0)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

67.45/38.87

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

Removed 1 trailing tuple parts

67.45/38.87

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, z0, s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

67.45/38.87

(21) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

COND(true, s(z0), 0) → c(COND(true, z0, s(0)))

Removed 1 trailing nodes:

COND(true, 0, z0) → c

67.45/38.87

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, 0, z0) → c

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

67.45/38.87

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

Use narrowing to replace COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1))) by

COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1))) 67.45/38.87
COND(true, s(0), s(z0)) → c(COND(false, p(s(0)), s(s(z0))), GR(s(0), s(z0))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

67.45/38.87

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, s(0), s(z0)) → c(COND(false, p(s(0)), s(s(z0))), GR(s(0), s(z0))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1))) 67.45/38.87
COND(true, s(0), s(z0)) → c(COND(false, p(s(0)), s(s(z0))), GR(s(0), s(z0))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, 0, z0) → c

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

67.45/38.87

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

Removed 1 trailing tuple parts

67.45/38.87

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, 0, z0) → c

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c

67.45/38.87

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

Split RHS of tuples not part of any SCC

67.45/38.87

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, 0, z0) → c

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c, c1

67.45/38.87

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

Removed 1 trailing nodes:

COND(true, 0, z0) → c

67.45/38.87

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, 0, z0) → c

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c, c1

67.45/38.87

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

The following tuples could be moved from S to K by knowledge propagation:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0))) 67.45/38.87
COND(true, 0, z0) → c

67.45/38.87

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c, c1

67.45/38.87

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

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

COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))

We considered the (Usable) Rules:

p(s(z0)) → z0 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1)

And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

The order we found is given by the following interpretation:
Polynomial interpretation : 67.45/38.87

POL(0) = [5] 67.45/38.87
POL(COND(x₁, x₂, x₃)) = [2] 67.45/38.87
POL(GR(x₁, x₂)) = 0 67.45/38.87
POL(c) = 0 67.45/38.87
POL(c(x₁)) = x₁ 67.45/38.87
POL(c(x₁, x₂)) = x₁ + x₂ 67.45/38.87
POL(c1(x₁)) = x₁ 67.45/38.87
POL(c3(x₁)) = x₁ 67.45/38.87
POL(false) = [5] 67.45/38.87
POL(gr(x₁, x₂)) = [5] + x₁ 67.45/38.87
POL(p(x₁)) = [1] 67.45/38.87
POL(s(x₁)) = 0 67.45/38.87
POL(true) = 0

67.45/38.87

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0))) 67.45/38.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c, c1

67.45/38.87

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

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

COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))

We considered the (Usable) Rules:

p(s(z0)) → z0 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1)

And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.87
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.87
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.87
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.87
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.87
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

The order we found is given by the following interpretation:
Polynomial interpretation : 67.45/38.87

POL(0) = [4] 67.45/38.87
POL(COND(x₁, x₂, x₃)) = [4] + x₂ 67.45/38.87
POL(GR(x₁, x₂)) = 0 67.45/38.87
POL(c) = 0 67.45/38.87
POL(c(x₁)) = x₁ 67.45/38.87
POL(c(x₁, x₂)) = x₁ + x₂ 67.45/38.87
POL(c1(x₁)) = x₁ 67.45/38.87
POL(c3(x₁)) = x₁ 67.45/38.87
POL(false) = 0 67.45/38.87
POL(gr(x₁, x₂)) = 0 67.45/38.87
POL(p(x₁)) = x₁ 67.45/38.87
POL(s(x₁)) = [2] + x₁ 67.45/38.87
POL(true) = 0

67.45/38.87

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.87
gr(0, z0) → false 67.45/38.87
gr(s(z0), 0) → true 67.45/38.87
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.87
p(0) → 0 67.45/38.87
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.87
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.87
COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.88
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.88
COND(true, 0, z0) → c 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c, c1

67.45/38.88

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

Use narrowing to replace COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) by

COND(true, 0, z0) → c(COND(false, 0, s(z0)))

67.45/38.88

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.88
gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(0) → 0 67.45/38.88
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.88
COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0))) 67.45/38.88
COND(true, 0, z0) → c(COND(false, 0, s(z0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
COND(true, 0, z0) → c(COND(false, 0, s(z0)))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.88
COND(true, 0, z0) → c 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c, c1

67.45/38.88

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

Removed 1 trailing tuple parts

67.45/38.88

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.88
gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(0) → 0 67.45/38.88
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.88
COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
COND(true, 0, z0) → c

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.88
COND(true, 0, z0) → c 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c, c1

67.45/38.88

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

Removed 1 trailing nodes:

COND(true, 0, z0) → c

67.45/38.88

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.88
gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(0) → 0 67.45/38.88
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.88
COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
COND(true, 0, z0) → c

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.88
COND(true, 0, z0) → c 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c, c1

67.45/38.88

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

The following tuples could be moved from S to K by knowledge propagation:

COND(true, 0, z0) → c 67.45/38.88
COND(true, 0, z0) → c

67.45/38.88

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.88
gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(0) → 0 67.45/38.88
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.88
COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.88
COND(true, 0, z0) → c 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c, c1

67.45/38.88

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

Use narrowing to replace COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) by

COND(true, s(z0), 0) → c(COND(true, z0, s(0)))

67.45/38.88

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.88
gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(0) → 0 67.45/38.88
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.88
COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0))) 67.45/38.88
COND(true, s(z0), 0) → c(COND(true, z0, s(0)))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) 67.45/38.88
COND(true, 0, z0) → c 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c, c1

67.45/38.88

(47) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

COND(true, s(z0), 0) → c(COND(true, z0, s(0))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

Removed 1 trailing nodes:

COND(true, 0, z0) → c

67.45/38.88

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.88
gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(0) → 0 67.45/38.88
p(s(z0)) → z0

Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.88
COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))

S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1)) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c, c1

67.45/38.88

(49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace GR(s(z0), s(z1)) → c3(GR(z0, z1)) by

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))

67.45/38.88

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.88
gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(0) → 0 67.45/38.88
p(s(z0)) → z0

Tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))

S tuples:

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))

K tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c, c1, c3

67.45/38.88

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

Removed 1 trailing tuple parts

67.45/38.88

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.88
gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(0) → 0 67.45/38.88
p(s(z0)) → z0

Tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 67.45/38.88
COND(true, s(0), s(z0)) → c

S tuples:

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))

K tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(0), s(z0)) → c

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c1, c3

67.45/38.88

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

Removed 2 trailing nodes:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(0), s(z0)) → c

67.45/38.88

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.88
gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(0) → 0 67.45/38.88
p(s(z0)) → z0

Tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 67.45/38.88
COND(true, s(0), s(z0)) → c

S tuples:

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))

K tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(0), s(z0)) → c

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c1, c3

67.45/38.88

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

Used rewriting to replace COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) by COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

67.45/38.88

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.88
gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(0) → 0 67.45/38.88
p(s(z0)) → z0

Tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 67.45/38.88
COND(true, s(0), s(z0)) → c 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

S tuples:

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(0), s(z0)) → c

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c1, c3

67.45/38.88

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

Removed 2 trailing nodes:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(0), s(z0)) → c

67.45/38.88

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.88
gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(0) → 0 67.45/38.88
p(s(z0)) → z0

Tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 67.45/38.88
COND(true, s(0), s(z0)) → c 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

S tuples:

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(0), s(z0)) → c

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c1, c3

67.45/38.88

(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.

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

We considered the (Usable) Rules:

gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(s(z0)) → z0

And the Tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 67.45/38.88
COND(true, s(0), s(z0)) → c 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

The order we found is given by the following interpretation:
Polynomial interpretation : 67.45/38.88

POL(0) = 0 67.45/38.88
POL(COND(x₁, x₂, x₃)) = [5]x₂ 67.45/38.88
POL(GR(x₁, x₂)) = 0 67.45/38.88
POL(c) = 0 67.45/38.88
POL(c(x₁, x₂)) = x₁ + x₂ 67.45/38.88
POL(c1(x₁)) = x₁ 67.45/38.88
POL(c3(x₁)) = x₁ 67.45/38.88
POL(false) = [3] 67.45/38.88
POL(gr(x₁, x₂)) = 0 67.45/38.88
POL(p(x₁)) = x₁ 67.45/38.88
POL(s(x₁)) = [2] + x₁ 67.45/38.88
POL(true) = 0

67.45/38.88

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.88
gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(0) → 0 67.45/38.88
p(s(z0)) → z0

Tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 67.45/38.88
COND(true, s(0), s(z0)) → c 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

S tuples:

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))

K tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(0), s(z0)) → c 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c1, c3

67.45/38.88

(61) 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.

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))

We considered the (Usable) Rules:

gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(s(z0)) → z0

And the Tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 67.45/38.88
COND(true, s(0), s(z0)) → c 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

The order we found is given by the following interpretation:
Polynomial interpretation : 67.45/38.88

POL(0) = 0 67.45/38.88
POL(COND(x₁, x₂, x₃)) = [2]x₂² 67.45/38.88
POL(GR(x₁, x₂)) = [2] + x₁ 67.45/38.88
POL(c) = 0 67.45/38.88
POL(c(x₁, x₂)) = x₁ + x₂ 67.45/38.88
POL(c1(x₁)) = x₁ 67.45/38.88
POL(c3(x₁)) = x₁ 67.45/38.88
POL(false) = [3] 67.45/38.88
POL(gr(x₁, x₂)) = 0 67.45/38.88
POL(p(x₁)) = x₁ 67.45/38.88
POL(s(x₁)) = [2] + x₁ 67.45/38.88
POL(true) = 0

67.45/38.88

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1)) 67.45/38.88
gr(0, z0) → false 67.45/38.88
gr(s(z0), 0) → true 67.45/38.88
gr(s(z0), s(z1)) → gr(z0, z1) 67.45/38.88
p(0) → 0 67.45/38.88
p(s(z0)) → z0

Tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1))) 67.45/38.88
COND(true, s(0), s(z0)) → c 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))

S tuples:none
K tuples:

COND(true, 0, z0) → c 67.45/38.88
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0)))) 67.45/38.88
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1))) 67.45/38.88
COND(true, s(0), s(z0)) → c 67.45/38.88
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) 67.45/38.88
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))

Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c1, c3

67.45/38.88

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

The set S is empty

67.45/38.88

(64) BOUNDS(O(1), O(1))

67.45/38.88