YES(O(1), O(n^2)) 19.29/8.60 YES(O(1), O(n^2)) 19.29/8.62 19.29/8.62 19.29/8.62 19.29/8.62 19.29/8.62 19.29/8.62 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 19.29/8.62 19.29/8.62 19.29/8.62
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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))
19.29/8.62 
↳2 CdtProblem
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↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
19.29/8.62 
↳4 CdtProblem
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↳5 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
19.29/8.62 
↳6 CdtProblem
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↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
19.29/8.62 
↳8 CdtProblem
19.29/8.62 
↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
19.29/8.62 
↳10 CdtProblem
19.29/8.62 
↳11 CdtLeafRemovalProof (ComplexityIfPolyImplication)
19.29/8.62 
↳12 CdtProblem
19.29/8.62 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
19.29/8.62 
↳14 CdtProblem
19.29/8.62 
↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
19.29/8.62 
↳16 CdtProblem
19.29/8.62 
↳17 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
19.29/8.62 
↳18 CdtProblem
19.29/8.62 
↳19 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
19.29/8.62 
↳20 CdtProblem
19.29/8.62 
↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
19.29/8.62 
↳22 CdtProblem
19.29/8.62 
↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
19.29/8.62 
↳24 CdtProblem
19.29/8.62 
↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
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↳26 CdtProblem
19.29/8.62 
↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
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↳28 CdtProblem
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↳29 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳30 BOUNDS(O(1), O(1))
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19.29/8.62

(0) Obligation:

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

lt(0, s(x)) → true 19.29/8.62
lt(x, 0) → false 19.29/8.62
lt(s(x), s(y)) → lt(x, y) 19.29/8.62
logarithm(x) → ifa(lt(0, x), x) 19.29/8.62
ifa(true, x) → help(x, 1) 19.29/8.62
ifa(false, x) → logZeroError 19.29/8.62
help(x, y) → ifb(lt(y, x), x, y) 19.29/8.62
ifb(true, x, y) → help(half(x), s(y)) 19.29/8.62
ifb(false, x, y) → y 19.29/8.62
half(0) → 0 19.29/8.62
half(s(0)) → 0 19.29/8.62
half(s(s(x))) → s(half(x))

Rewrite Strategy: INNERMOST
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19.29/8.62

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

Converted CpxTRS to CDT
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19.29/8.62

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 19.29/8.62
lt(z0, 0) → false 19.29/8.62
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.62
logarithm(z0) → ifa(lt(0, z0), z0) 19.29/8.62
ifa(true, z0) → help(z0, 1) 19.29/8.62
ifa(false, z0) → logZeroError 19.29/8.62
help(z0, z1) → ifb(lt(z1, z0), z0, z1) 19.29/8.62
ifb(true, z0, z1) → help(half(z0), s(z1)) 19.29/8.62
ifb(false, z0, z1) → z1 19.29/8.62
half(0) → 0 19.29/8.62
half(s(0)) → 0 19.29/8.62
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.62
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0), LT(0, z0)) 19.29/8.62
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.62
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0)) 19.29/8.62
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.62
HALF(s(s(z0))) → c11(HALF(z0))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.62
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0), LT(0, z0)) 19.29/8.62
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.62
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0)) 19.29/8.62
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.62
HALF(s(s(z0))) → c11(HALF(z0))
K tuples:none
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, LOGARITHM, IFA, HELP, IFB, HALF

Compound Symbols:

c2, c3, c4, c6, c7, c11

19.29/8.62
19.29/8.62

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

Removed 1 trailing tuple parts
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19.29/8.62

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 19.29/8.62
lt(z0, 0) → false 19.29/8.62
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.62
logarithm(z0) → ifa(lt(0, z0), z0) 19.29/8.62
ifa(true, z0) → help(z0, 1) 19.29/8.62
ifa(false, z0) → logZeroError 19.29/8.64
help(z0, z1) → ifb(lt(z1, z0), z0, z1) 19.29/8.64
ifb(true, z0, z1) → help(half(z0), s(z1)) 19.29/8.64
ifb(false, z0, z1) → z1 19.29/8.64
half(0) → 0 19.29/8.64
half(s(0)) → 0 19.29/8.64
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.64
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.64
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0)) 19.29/8.64
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.64
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.64
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.64
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.64
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0)) 19.29/8.64
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.64
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.64
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0))
K tuples:none
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, HELP, IFB, HALF, LOGARITHM

Compound Symbols:

c2, c4, c6, c7, c11, c3

19.29/8.64
19.29/8.64

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

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

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.64
IFA(true, z0) → c4(HELP(z0, 1))
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19.29/8.64

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 19.29/8.64
lt(z0, 0) → false 19.29/8.64
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.64
logarithm(z0) → ifa(lt(0, z0), z0) 19.29/8.64
ifa(true, z0) → help(z0, 1) 19.29/8.64
ifa(false, z0) → logZeroError 19.29/8.64
help(z0, z1) → ifb(lt(z1, z0), z0, z1) 19.29/8.64
ifb(true, z0, z1) → help(half(z0), s(z1)) 19.29/8.64
ifb(false, z0, z1) → z1 19.29/8.64
half(0) → 0 19.29/8.64
half(s(0)) → 0 19.29/8.64
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.64
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.64
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0)) 19.29/8.64
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.64
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.64
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.64
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0)) 19.29/8.64
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.64
HALF(s(s(z0))) → c11(HALF(z0))
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.64
IFA(true, z0) → c4(HELP(z0, 1))
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, HELP, IFB, HALF, LOGARITHM

Compound Symbols:

c2, c4, c6, c7, c11, c3

19.29/8.64
19.29/8.64

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

Use narrowing to replace HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0)) by

HELP(s(z0), 0) → c6(IFB(true, s(z0), 0), LT(0, s(z0))) 19.29/8.64
HELP(0, z0) → c6(IFB(false, 0, z0), LT(z0, 0)) 19.29/8.64
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
19.29/8.64
19.29/8.64

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 19.29/8.64
lt(z0, 0) → false 19.29/8.64
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.64
logarithm(z0) → ifa(lt(0, z0), z0) 19.29/8.64
ifa(true, z0) → help(z0, 1) 19.29/8.64
ifa(false, z0) → logZeroError 19.29/8.64
help(z0, z1) → ifb(lt(z1, z0), z0, z1) 19.29/8.64
ifb(true, z0, z1) → help(half(z0), s(z1)) 19.29/8.64
ifb(false, z0, z1) → z1 19.29/8.64
half(0) → 0 19.29/8.64
half(s(0)) → 0 19.29/8.64
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.64
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.64
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.64
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.64
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.64
HELP(s(z0), 0) → c6(IFB(true, s(z0), 0), LT(0, s(z0))) 19.29/8.64
HELP(0, z0) → c6(IFB(false, 0, z0), LT(z0, 0)) 19.29/8.64
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.64
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.64
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.64
HELP(s(z0), 0) → c6(IFB(true, s(z0), 0), LT(0, s(z0))) 19.29/8.64
HELP(0, z0) → c6(IFB(false, 0, z0), LT(z0, 0)) 19.29/8.64
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.64
IFA(true, z0) → c4(HELP(z0, 1))
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, IFB, HALF, LOGARITHM, HELP

Compound Symbols:

c2, c4, c7, c11, c3, c6

19.29/8.64
19.29/8.64

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

Removed 3 trailing tuple parts
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19.29/8.64

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 19.29/8.64
lt(z0, 0) → false 19.29/8.64
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.64
logarithm(z0) → ifa(lt(0, z0), z0) 19.29/8.64
ifa(true, z0) → help(z0, 1) 19.29/8.64
ifa(false, z0) → logZeroError 19.29/8.64
help(z0, z1) → ifb(lt(z1, z0), z0, z1) 19.29/8.64
ifb(true, z0, z1) → help(half(z0), s(z1)) 19.29/8.64
ifb(false, z0, z1) → z1 19.29/8.64
half(0) → 0 19.29/8.64
half(s(0)) → 0 19.29/8.64
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.64
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.64
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.64
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.64
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.64
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.64
HELP(s(z0), 0) → c6(IFB(true, s(z0), 0)) 19.29/8.64
HELP(0, z0) → c6
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.64
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.64
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.64
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.64
HELP(s(z0), 0) → c6(IFB(true, s(z0), 0)) 19.29/8.64
HELP(0, z0) → c6
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.64
IFA(true, z0) → c4(HELP(z0, 1))
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, IFB, HALF, LOGARITHM, HELP

Compound Symbols:

c2, c4, c7, c11, c3, c6, c6, c6

19.29/8.64
19.29/8.64

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

HELP(s(z0), 0) → c6(IFB(true, s(z0), 0))
Removed 3 trailing nodes:

HELP(0, z0) → c6 19.29/8.64
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.64
IFA(true, z0) → c4(HELP(z0, 1))
19.29/8.64
19.29/8.64

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 19.29/8.64
lt(z0, 0) → false 19.29/8.64
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.64
logarithm(z0) → ifa(lt(0, z0), z0) 19.29/8.64
ifa(true, z0) → help(z0, 1) 19.29/8.64
ifa(false, z0) → logZeroError 19.29/8.64
help(z0, z1) → ifb(lt(z1, z0), z0, z1) 19.29/8.64
ifb(true, z0, z1) → help(half(z0), s(z1)) 19.29/8.64
ifb(false, z0, z1) → z1 19.29/8.64
half(0) → 0 19.29/8.64
half(s(0)) → 0 19.29/8.64
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.64
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.64
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.64
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.64
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.64
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.64
HELP(0, z0) → c6
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.64
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.64
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.64
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.64
HELP(0, z0) → c6
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.64
IFA(true, z0) → c4(HELP(z0, 1))
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, IFB, HALF, LOGARITHM, HELP

Compound Symbols:

c2, c4, c7, c11, c3, c6, c6

19.29/8.64
19.29/8.64

(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.

HELP(0, z0) → c6
We considered the (Usable) Rules:

lt(0, s(z0)) → true 19.29/8.64
lt(z0, 0) → false 19.29/8.64
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.64
half(0) → 0 19.29/8.64
half(s(0)) → 0 19.29/8.64
half(s(s(z0))) → s(half(z0))
And the Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.64
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.64
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.64
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.64
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.64
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.64
HELP(0, z0) → c6
The order we found is given by the following interpretation:
Polynomial interpretation : 19.29/8.64

POL(0) = [1]    19.29/8.64
POL(1) = [1]    19.29/8.64
POL(HALF(x1)) = 0    19.29/8.64
POL(HELP(x1, x2)) = x1    19.29/8.64
POL(IFA(x1, x2)) = [5]x2    19.29/8.64
POL(IFB(x1, x2, x3)) = x2    19.29/8.64
POL(LOGARITHM(x1)) = [4] + [5]x1    19.29/8.64
POL(LT(x1, x2)) = 0    19.29/8.64
POL(c11(x1)) = x1    19.29/8.64
POL(c2(x1)) = x1    19.29/8.66
POL(c3(x1)) = x1    19.29/8.66
POL(c4(x1)) = x1    19.29/8.66
POL(c6) = 0    19.29/8.66
POL(c6(x1, x2)) = x1 + x2    19.29/8.66
POL(c7(x1, x2)) = x1 + x2    19.29/8.66
POL(false) = [3]    19.29/8.66
POL(half(x1)) = x1    19.29/8.66
POL(lt(x1, x2)) = 0    19.29/8.66
POL(s(x1)) = x1    19.29/8.66
POL(true) = 0   
19.29/8.66
19.29/8.66

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 19.29/8.66
lt(z0, 0) → false 19.29/8.66
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.66
logarithm(z0) → ifa(lt(0, z0), z0) 19.29/8.66
ifa(true, z0) → help(z0, 1) 19.29/8.66
ifa(false, z0) → logZeroError 19.29/8.66
help(z0, z1) → ifb(lt(z1, z0), z0, z1) 19.29/8.66
ifb(true, z0, z1) → help(half(z0), s(z1)) 19.29/8.66
ifb(false, z0, z1) → z1 19.29/8.66
half(0) → 0 19.29/8.66
half(s(0)) → 0 19.29/8.66
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.66
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
HELP(0, z0) → c6
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
HELP(0, z0) → c6
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, IFB, HALF, LOGARITHM, HELP

Compound Symbols:

c2, c4, c7, c11, c3, c6, c6

19.29/8.66
19.29/8.66

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

Use narrowing to replace IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) by

IFB(true, 0, x1) → c7(HELP(0, s(x1)), HALF(0)) 19.29/8.66
IFB(true, s(0), x1) → c7(HELP(0, s(x1)), HALF(s(0))) 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
19.29/8.66
19.29/8.66

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 19.29/8.66
lt(z0, 0) → false 19.29/8.66
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.66
logarithm(z0) → ifa(lt(0, z0), z0) 19.29/8.66
ifa(true, z0) → help(z0, 1) 19.29/8.66
ifa(false, z0) → logZeroError 19.29/8.66
help(z0, z1) → ifb(lt(z1, z0), z0, z1) 19.29/8.66
ifb(true, z0, z1) → help(half(z0), s(z1)) 19.29/8.66
ifb(false, z0, z1) → z1 19.29/8.66
half(0) → 0 19.29/8.66
half(s(0)) → 0 19.29/8.66
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.66
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
HELP(0, z0) → c6 19.29/8.66
IFB(true, 0, x1) → c7(HELP(0, s(x1)), HALF(0)) 19.29/8.66
IFB(true, s(0), x1) → c7(HELP(0, s(x1)), HALF(s(0))) 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
IFB(true, 0, x1) → c7(HELP(0, s(x1)), HALF(0)) 19.29/8.66
IFB(true, s(0), x1) → c7(HELP(0, s(x1)), HALF(s(0))) 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
HELP(0, z0) → c6
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, HALF, LOGARITHM, HELP, IFB

Compound Symbols:

c2, c4, c11, c3, c6, c6, c7

19.29/8.66
19.29/8.66

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

Removed 2 trailing tuple parts
19.29/8.66
19.29/8.66

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 19.29/8.66
lt(z0, 0) → false 19.29/8.66
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.66
logarithm(z0) → ifa(lt(0, z0), z0) 19.29/8.66
ifa(true, z0) → help(z0, 1) 19.29/8.66
ifa(false, z0) → logZeroError 19.29/8.66
help(z0, z1) → ifb(lt(z1, z0), z0, z1) 19.29/8.66
ifb(true, z0, z1) → help(half(z0), s(z1)) 19.29/8.66
ifb(false, z0, z1) → z1 19.29/8.66
half(0) → 0 19.29/8.66
half(s(0)) → 0 19.29/8.66
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.66
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
HELP(0, z0) → c6 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0)))) 19.29/8.66
IFB(true, 0, x1) → c7(HELP(0, s(x1))) 19.29/8.66
IFB(true, s(0), x1) → c7(HELP(0, s(x1)))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0)))) 19.29/8.66
IFB(true, 0, x1) → c7(HELP(0, s(x1))) 19.29/8.66
IFB(true, s(0), x1) → c7(HELP(0, s(x1)))
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
HELP(0, z0) → c6
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, HALF, LOGARITHM, HELP, IFB

Compound Symbols:

c2, c4, c11, c3, c6, c6, c7, c7

19.29/8.66
19.29/8.66

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

Removed 5 trailing nodes:

IFB(true, s(0), x1) → c7(HELP(0, s(x1))) 19.29/8.66
IFB(true, 0, x1) → c7(HELP(0, s(x1))) 19.29/8.66
HELP(0, z0) → c6 19.29/8.66
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1))
19.29/8.66
19.29/8.66

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 19.29/8.66
lt(z0, 0) → false 19.29/8.66
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.66
logarithm(z0) → ifa(lt(0, z0), z0) 19.29/8.66
ifa(true, z0) → help(z0, 1) 19.29/8.66
ifa(false, z0) → logZeroError 19.29/8.66
help(z0, z1) → ifb(lt(z1, z0), z0, z1) 19.29/8.66
ifb(true, z0, z1) → help(half(z0), s(z1)) 19.29/8.66
ifb(false, z0, z1) → z1 19.29/8.66
half(0) → 0 19.29/8.66
half(s(0)) → 0 19.29/8.66
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.66
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
HELP(0, z0) → c6 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0)))) 19.29/8.66
IFB(true, s(0), x1) → c7(HELP(0, s(x1)))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0)))) 19.29/8.66
IFB(true, s(0), x1) → c7(HELP(0, s(x1)))
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
HELP(0, z0) → c6
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, HALF, LOGARITHM, HELP, IFB

Compound Symbols:

c2, c4, c11, c3, c6, c6, c7, c7

19.29/8.66
19.29/8.66

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

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

IFB(true, s(0), x1) → c7(HELP(0, s(x1)))
We considered the (Usable) Rules:

half(0) → 0 19.29/8.66
half(s(0)) → 0 19.29/8.66
half(s(s(z0))) → s(half(z0)) 19.29/8.66
lt(0, s(z0)) → true 19.29/8.66
lt(z0, 0) → false 19.29/8.66
lt(s(z0), s(z1)) → lt(z0, z1)
And the Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.66
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
HELP(0, z0) → c6 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0)))) 19.29/8.66
IFB(true, s(0), x1) → c7(HELP(0, s(x1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 19.29/8.66

POL(0) = [3]    19.29/8.66
POL(1) = [1]    19.29/8.66
POL(HALF(x1)) = 0    19.29/8.66
POL(HELP(x1, x2)) = [1] + [2]x1    19.29/8.66
POL(IFA(x1, x2)) = [1] + [3]x2    19.29/8.66
POL(IFB(x1, x2, x3)) = [1] + [2]x2    19.29/8.66
POL(LOGARITHM(x1)) = [3] + [3]x1    19.29/8.66
POL(LT(x1, x2)) = 0    19.29/8.66
POL(c11(x1)) = x1    19.29/8.66
POL(c2(x1)) = x1    19.29/8.66
POL(c3(x1)) = x1    19.29/8.66
POL(c4(x1)) = x1    19.29/8.66
POL(c6) = 0    19.29/8.66
POL(c6(x1, x2)) = x1 + x2    19.29/8.66
POL(c7(x1)) = x1    19.29/8.66
POL(c7(x1, x2)) = x1 + x2    19.29/8.66
POL(false) = [3]    19.29/8.66
POL(half(x1)) = 0    19.29/8.66
POL(lt(x1, x2)) = 0    19.29/8.66
POL(s(x1)) = [4]    19.29/8.66
POL(true) = 0   
19.29/8.66
19.29/8.66

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 19.29/8.66
lt(z0, 0) → false 19.29/8.66
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.66
logarithm(z0) → ifa(lt(0, z0), z0) 19.29/8.66
ifa(true, z0) → help(z0, 1) 19.29/8.66
ifa(false, z0) → logZeroError 19.29/8.66
help(z0, z1) → ifb(lt(z1, z0), z0, z1) 19.29/8.66
ifb(true, z0, z1) → help(half(z0), s(z1)) 19.29/8.66
ifb(false, z0, z1) → z1 19.29/8.66
half(0) → 0 19.29/8.66
half(s(0)) → 0 19.29/8.66
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.66
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
HELP(0, z0) → c6 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0)))) 19.29/8.66
IFB(true, s(0), x1) → c7(HELP(0, s(x1)))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
HELP(0, z0) → c6 19.29/8.66
IFB(true, s(0), x1) → c7(HELP(0, s(x1)))
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, HALF, LOGARITHM, HELP, IFB

Compound Symbols:

c2, c4, c11, c3, c6, c6, c7, c7

19.29/8.66
19.29/8.66

(23) 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.

HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
We considered the (Usable) Rules:

half(0) → 0 19.29/8.66
half(s(0)) → 0 19.29/8.66
half(s(s(z0))) → s(half(z0)) 19.29/8.66
lt(0, s(z0)) → true 19.29/8.66
lt(z0, 0) → false 19.29/8.66
lt(s(z0), s(z1)) → lt(z0, z1)
And the Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.66
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
HELP(0, z0) → c6 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0)))) 19.29/8.66
IFB(true, s(0), x1) → c7(HELP(0, s(x1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 19.29/8.66

POL(0) = 0    19.29/8.66
POL(1) = 0    19.29/8.66
POL(HALF(x1)) = 0    19.29/8.66
POL(HELP(x1, x2)) = [2] + [4]x1 + [3]x2    19.29/8.66
POL(IFA(x1, x2)) = [4] + [5]x2    19.29/8.66
POL(IFB(x1, x2, x3)) = [1] + [4]x2 + [3]x3    19.29/8.66
POL(LOGARITHM(x1)) = [4] + [5]x1    19.29/8.66
POL(LT(x1, x2)) = 0    19.29/8.66
POL(c11(x1)) = x1    19.29/8.66
POL(c2(x1)) = x1    19.29/8.66
POL(c3(x1)) = x1    19.29/8.66
POL(c4(x1)) = x1    19.29/8.66
POL(c6) = 0    19.29/8.66
POL(c6(x1, x2)) = x1 + x2    19.29/8.66
POL(c7(x1)) = x1    19.29/8.66
POL(c7(x1, x2)) = x1 + x2    19.29/8.66
POL(false) = [3]    19.29/8.66
POL(half(x1)) = x1    19.29/8.66
POL(lt(x1, x2)) = 0    19.29/8.66
POL(s(x1)) = [2] + x1    19.29/8.66
POL(true) = 0   
19.29/8.66
19.29/8.66

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 19.29/8.66
lt(z0, 0) → false 19.29/8.66
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.66
logarithm(z0) → ifa(lt(0, z0), z0) 19.29/8.66
ifa(true, z0) → help(z0, 1) 19.29/8.66
ifa(false, z0) → logZeroError 19.29/8.66
help(z0, z1) → ifb(lt(z1, z0), z0, z1) 19.29/8.66
ifb(true, z0, z1) → help(half(z0), s(z1)) 19.29/8.66
ifb(false, z0, z1) → z1 19.29/8.66
half(0) → 0 19.29/8.66
half(s(0)) → 0 19.29/8.66
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.66
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
HELP(0, z0) → c6 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0)))) 19.29/8.66
IFB(true, s(0), x1) → c7(HELP(0, s(x1)))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0))
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
HELP(0, z0) → c6 19.29/8.66
IFB(true, s(0), x1) → c7(HELP(0, s(x1))) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, HALF, LOGARITHM, HELP, IFB

Compound Symbols:

c2, c4, c11, c3, c6, c6, c7, c7

19.29/8.66
19.29/8.66

(25) 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.

HALF(s(s(z0))) → c11(HALF(z0))
We considered the (Usable) Rules:

half(0) → 0 19.29/8.66
half(s(0)) → 0 19.29/8.66
half(s(s(z0))) → s(half(z0)) 19.29/8.66
lt(0, s(z0)) → true 19.29/8.66
lt(z0, 0) → false 19.29/8.66
lt(s(z0), s(z1)) → lt(z0, z1)
And the Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.66
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.66
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.66
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.66
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.66
HELP(0, z0) → c6 19.29/8.66
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0)))) 19.29/8.66
IFB(true, s(0), x1) → c7(HELP(0, s(x1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 19.29/8.66

POL(0) = 0    19.29/8.66
POL(1) = 0    19.29/8.66
POL(HALF(x1)) = x1    19.29/8.66
POL(HELP(x1, x2)) = [2] + [2]x12    19.29/8.66
POL(IFA(x1, x2)) = [2] + [2]x22    19.29/8.66
POL(IFB(x1, x2, x3)) = [2]x22    19.29/8.66
POL(LOGARITHM(x1)) = [3] + x1 + [2]x12    19.29/8.66
POL(LT(x1, x2)) = 0    19.29/8.66
POL(c11(x1)) = x1    19.29/8.66
POL(c2(x1)) = x1    19.29/8.66
POL(c3(x1)) = x1    19.29/8.66
POL(c4(x1)) = x1    19.29/8.66
POL(c6) = 0    19.29/8.66
POL(c6(x1, x2)) = x1 + x2    19.29/8.66
POL(c7(x1)) = x1    19.29/8.66
POL(c7(x1, x2)) = x1 + x2    19.29/8.66
POL(false) = [3]    19.29/8.66
POL(half(x1)) = x1    19.29/8.67
POL(lt(x1, x2)) = 0    19.29/8.67
POL(s(x1)) = [1] + x1    19.29/8.67
POL(true) = 0   
19.29/8.67
19.29/8.67

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 19.29/8.67
lt(z0, 0) → false 19.29/8.67
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.67
logarithm(z0) → ifa(lt(0, z0), z0) 19.29/8.67
ifa(true, z0) → help(z0, 1) 19.29/8.67
ifa(false, z0) → logZeroError 19.29/8.67
help(z0, z1) → ifb(lt(z1, z0), z0, z1) 19.29/8.67
ifb(true, z0, z1) → help(half(z0), s(z1)) 19.29/8.67
ifb(false, z0, z1) → z1 19.29/8.67
half(0) → 0 19.29/8.67
half(s(0)) → 0 19.29/8.67
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.67
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.67
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.67
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.67
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.67
HELP(0, z0) → c6 19.29/8.67
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0)))) 19.29/8.67
IFB(true, s(0), x1) → c7(HELP(0, s(x1)))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.67
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.67
HELP(0, z0) → c6 19.29/8.67
IFB(true, s(0), x1) → c7(HELP(0, s(x1))) 19.29/8.67
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.67
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0)))) 19.29/8.67
HALF(s(s(z0))) → c11(HALF(z0))
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, HALF, LOGARITHM, HELP, IFB

Compound Symbols:

c2, c4, c11, c3, c6, c6, c7, c7

19.29/8.67
19.29/8.67

(27) 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.

LT(s(z0), s(z1)) → c2(LT(z0, z1))
We considered the (Usable) Rules:

half(0) → 0 19.29/8.67
half(s(0)) → 0 19.29/8.67
half(s(s(z0))) → s(half(z0)) 19.29/8.67
lt(0, s(z0)) → true 19.29/8.67
lt(z0, 0) → false 19.29/8.67
lt(s(z0), s(z1)) → lt(z0, z1)
And the Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.67
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.67
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.67
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.67
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.67
HELP(0, z0) → c6 19.29/8.67
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0)))) 19.29/8.67
IFB(true, s(0), x1) → c7(HELP(0, s(x1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 19.29/8.67

POL(0) = 0    19.29/8.67
POL(1) = 0    19.29/8.67
POL(HALF(x1)) = 0    19.29/8.67
POL(HELP(x1, x2)) = x1 + x12    19.29/8.67
POL(IFA(x1, x2)) = x2 + [2]x22    19.29/8.67
POL(IFB(x1, x2, x3)) = x22    19.29/8.67
POL(LOGARITHM(x1)) = x1 + [2]x12    19.29/8.67
POL(LT(x1, x2)) = x2    19.29/8.67
POL(c11(x1)) = x1    19.29/8.67
POL(c2(x1)) = x1    19.29/8.67
POL(c3(x1)) = x1    19.29/8.67
POL(c4(x1)) = x1    19.29/8.67
POL(c6) = 0    19.29/8.67
POL(c6(x1, x2)) = x1 + x2    19.29/8.67
POL(c7(x1)) = x1    19.29/8.67
POL(c7(x1, x2)) = x1 + x2    19.29/8.67
POL(false) = [3]    19.29/8.67
POL(half(x1)) = x1    19.29/8.67
POL(lt(x1, x2)) = 0    19.29/8.67
POL(s(x1)) = [2] + x1    19.29/8.67
POL(true) = 0   
19.29/8.67
19.29/8.67

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 19.29/8.67
lt(z0, 0) → false 19.29/8.67
lt(s(z0), s(z1)) → lt(z0, z1) 19.29/8.67
logarithm(z0) → ifa(lt(0, z0), z0) 19.29/8.67
ifa(true, z0) → help(z0, 1) 19.29/8.67
ifa(false, z0) → logZeroError 19.29/8.67
help(z0, z1) → ifb(lt(z1, z0), z0, z1) 19.29/8.67
ifb(true, z0, z1) → help(half(z0), s(z1)) 19.29/8.67
ifb(false, z0, z1) → z1 19.29/8.67
half(0) → 0 19.29/8.67
half(s(0)) → 0 19.29/8.67
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 19.29/8.67
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.67
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.67
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.67
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.67
HELP(0, z0) → c6 19.29/8.67
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0)))) 19.29/8.67
IFB(true, s(0), x1) → c7(HELP(0, s(x1)))
S tuples:none
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0)) 19.29/8.67
IFA(true, z0) → c4(HELP(z0, 1)) 19.29/8.67
HELP(0, z0) → c6 19.29/8.67
IFB(true, s(0), x1) → c7(HELP(0, s(x1))) 19.29/8.67
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1))) 19.29/8.67
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0)))) 19.29/8.67
HALF(s(s(z0))) → c11(HALF(z0)) 19.29/8.67
LT(s(z0), s(z1)) → c2(LT(z0, z1))
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, HALF, LOGARITHM, HELP, IFB

Compound Symbols:

c2, c4, c11, c3, c6, c6, c7, c7

19.29/8.67
19.29/8.67

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

The set S is empty
19.29/8.67
19.29/8.67

(30) BOUNDS(O(1), O(1))

19.29/8.67
19.29/8.67
19.29/8.69 EOF