YES(O(1), O(n^1)) 0.00/0.73 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 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳4 CdtProblem
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↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication)
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↳6 CdtProblem
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↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳8 CdtProblem
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↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳10 BOUNDS(O(1), O(1))
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(0) Obligation:

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

decrease(Cons(x, xs)) → decrease(xs) 0.00/0.75
decrease(Nil) → number42(Nil) 0.00/0.75
number42(x) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) 0.00/0.75
goal(x) → decrease(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:

decrease(Cons(z0, z1)) → decrease(z1) 0.00/0.75
decrease(Nil) → number42(Nil) 0.00/0.75
number42(z0) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) 0.00/0.75
goal(z0) → decrease(z0)
Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1)) 0.00/0.75
DECREASE(Nil) → c1(NUMBER42(Nil)) 0.00/0.75
GOAL(z0) → c3(DECREASE(z0))
S tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1)) 0.00/0.75
DECREASE(Nil) → c1(NUMBER42(Nil)) 0.00/0.75
GOAL(z0) → c3(DECREASE(z0))
K tuples:none
Defined Rule Symbols:

decrease, number42, goal

Defined Pair Symbols:

DECREASE, GOAL

Compound Symbols:

c, c1, 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:

decrease(Cons(z0, z1)) → decrease(z1) 0.00/0.75
decrease(Nil) → number42(Nil) 0.00/0.75
number42(z0) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) 0.00/0.75
goal(z0) → decrease(z0)
Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1)) 0.00/0.75
GOAL(z0) → c3(DECREASE(z0)) 0.00/0.75
DECREASE(Nil) → c1
S tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1)) 0.00/0.75
GOAL(z0) → c3(DECREASE(z0)) 0.00/0.75
DECREASE(Nil) → c1
K tuples:none
Defined Rule Symbols:

decrease, number42, goal

Defined Pair Symbols:

DECREASE, GOAL

Compound Symbols:

c, c3, c1

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(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

GOAL(z0) → c3(DECREASE(z0))
Removed 1 trailing nodes:

DECREASE(Nil) → c1
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

decrease(Cons(z0, z1)) → decrease(z1) 0.00/0.75
decrease(Nil) → number42(Nil) 0.00/0.75
number42(z0) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) 0.00/0.75
goal(z0) → decrease(z0)
Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1)) 0.00/0.75
DECREASE(Nil) → c1
S tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1)) 0.00/0.75
DECREASE(Nil) → c1
K tuples:none
Defined Rule Symbols:

decrease, number42, goal

Defined Pair Symbols:

DECREASE

Compound Symbols:

c, c1

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(7) 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.

DECREASE(Cons(z0, z1)) → c(DECREASE(z1)) 0.00/0.75
DECREASE(Nil) → c1
We considered the (Usable) Rules:none
And the Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1)) 0.00/0.75
DECREASE(Nil) → c1
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.75

POL(Cons(x1, x2)) = [3] + x2    0.00/0.75
POL(DECREASE(x1)) = [3] + [3]x1    0.00/0.75
POL(Nil) = [5]    0.00/0.75
POL(c(x1)) = x1    0.00/0.75
POL(c1) = 0   
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

decrease(Cons(z0, z1)) → decrease(z1) 0.00/0.75
decrease(Nil) → number42(Nil) 0.00/0.75
number42(z0) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) 0.00/0.75
goal(z0) → decrease(z0)
Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1)) 0.00/0.75
DECREASE(Nil) → c1
S tuples:none
K tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1)) 0.00/0.75
DECREASE(Nil) → c1
Defined Rule Symbols:

decrease, number42, goal

Defined Pair Symbols:

DECREASE

Compound Symbols:

c, c1

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

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

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0.00/0.78 EOF