YES(O(1), O(n^1)) 0.00/0.74 YES(O(1), O(n^1)) 0.00/0.75 0.00/0.75 0.00/0.75 0.00/0.75 0.00/0.75 0.00/0.75 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.75 0.00/0.75 0.00/0.75
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
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↳2 CdtProblem
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↳3 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳4 CdtProblem
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↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳6 CdtProblem
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↳7 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳8 BOUNDS(O(1), O(1))
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0.00/0.75 0.00/0.75
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0.00/0.75

(0) Obligation:

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

sum(0) → 0 0.00/0.75
sum(s(x)) → +(sum(x), s(x)) 0.00/0.75
sum1(0) → 0 0.00/0.75
sum1(s(x)) → s(+(sum1(x), +(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:

sum(0) → 0 0.00/0.75
sum(s(z0)) → +(sum(z0), s(z0)) 0.00/0.75
sum1(0) → 0 0.00/0.75
sum1(s(z0)) → s(+(sum1(z0), +(z0, z0)))
Tuples:

SUM(s(z0)) → c1(SUM(z0)) 0.00/0.75
SUM1(s(z0)) → c3(SUM1(z0))
S tuples:

SUM(s(z0)) → c1(SUM(z0)) 0.00/0.75
SUM1(s(z0)) → c3(SUM1(z0))
K tuples:none
Defined Rule Symbols:

sum, sum1

Defined Pair Symbols:

SUM, SUM1

Compound Symbols:

c1, c3

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(3) 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.

SUM1(s(z0)) → c3(SUM1(z0))
We considered the (Usable) Rules:none
And the Tuples:

SUM(s(z0)) → c1(SUM(z0)) 0.00/0.75
SUM1(s(z0)) → c3(SUM1(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.75

POL(SUM(x1)) = 0    0.00/0.75
POL(SUM1(x1)) = x1    0.00/0.75
POL(c1(x1)) = x1    0.00/0.75
POL(c3(x1)) = x1    0.00/0.75
POL(s(x1)) = [1] + x1   
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

sum(0) → 0 0.00/0.75
sum(s(z0)) → +(sum(z0), s(z0)) 0.00/0.75
sum1(0) → 0 0.00/0.75
sum1(s(z0)) → s(+(sum1(z0), +(z0, z0)))
Tuples:

SUM(s(z0)) → c1(SUM(z0)) 0.00/0.75
SUM1(s(z0)) → c3(SUM1(z0))
S tuples:

SUM(s(z0)) → c1(SUM(z0))
K tuples:

SUM1(s(z0)) → c3(SUM1(z0))
Defined Rule Symbols:

sum, sum1

Defined Pair Symbols:

SUM, SUM1

Compound Symbols:

c1, c3

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(5) 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.

SUM(s(z0)) → c1(SUM(z0))
We considered the (Usable) Rules:none
And the Tuples:

SUM(s(z0)) → c1(SUM(z0)) 0.00/0.75
SUM1(s(z0)) → c3(SUM1(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.75

POL(SUM(x1)) = [5]x1    0.00/0.75
POL(SUM1(x1)) = [5]x1    0.00/0.75
POL(c1(x1)) = x1    0.00/0.75
POL(c3(x1)) = x1    0.00/0.75
POL(s(x1)) = [1] + x1   
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0.00/0.75

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

sum(0) → 0 0.00/0.75
sum(s(z0)) → +(sum(z0), s(z0)) 0.00/0.75
sum1(0) → 0 0.00/0.75
sum1(s(z0)) → s(+(sum1(z0), +(z0, z0)))
Tuples:

SUM(s(z0)) → c1(SUM(z0)) 0.00/0.75
SUM1(s(z0)) → c3(SUM1(z0))
S tuples:none
K tuples:

SUM1(s(z0)) → c3(SUM1(z0)) 0.00/0.75
SUM(s(z0)) → c1(SUM(z0))
Defined Rule Symbols:

sum, sum1

Defined Pair Symbols:

SUM, SUM1

Compound Symbols:

c1, c3

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

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

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0.00/0.78 EOF