YES(O(1), O(n^1)) 0.00/0.70 YES(O(1), O(n^1)) 0.00/0.71 0.00/0.71 0.00/0.71 0.00/0.71 0.00/0.71 0.00/0.71 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.71 0.00/0.71 0.00/0.71
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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 CdtLeafRemovalProof (ComplexityIfPolyImplication)
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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) Obligation:

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

list(Cons(x, xs)) → list(xs) 0.00/0.71
list(Nil) → True 0.00/0.71
list(Nil) → isEmpty[Match](Nil) 0.00/0.71
notEmpty(Cons(x, xs)) → True 0.00/0.71
notEmpty(Nil) → False 0.00/0.71
goal(x) → list(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:

list(Cons(z0, z1)) → list(z1) 0.00/0.71
list(Nil) → True 0.00/0.71
list(Nil) → isEmpty[Match](Nil) 0.00/0.71
notEmpty(Cons(z0, z1)) → True 0.00/0.71
notEmpty(Nil) → False 0.00/0.71
goal(z0) → list(z0)
Tuples:

LIST(Cons(z0, z1)) → c(LIST(z1)) 0.00/0.71
GOAL(z0) → c5(LIST(z0))
S tuples:

LIST(Cons(z0, z1)) → c(LIST(z1)) 0.00/0.71
GOAL(z0) → c5(LIST(z0))
K tuples:none
Defined Rule Symbols:

list, notEmpty, goal

Defined Pair Symbols:

LIST, GOAL

Compound Symbols:

c, c5

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(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

GOAL(z0) → c5(LIST(z0))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

list(Cons(z0, z1)) → list(z1) 0.00/0.71
list(Nil) → True 0.00/0.71
list(Nil) → isEmpty[Match](Nil) 0.00/0.71
notEmpty(Cons(z0, z1)) → True 0.00/0.71
notEmpty(Nil) → False 0.00/0.71
goal(z0) → list(z0)
Tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))
S tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))
K tuples:none
Defined Rule Symbols:

list, notEmpty, goal

Defined Pair Symbols:

LIST

Compound Symbols:

c

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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.

LIST(Cons(z0, z1)) → c(LIST(z1))
We considered the (Usable) Rules:none
And the Tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.71

POL(Cons(x1, x2)) = [1] + x2    0.00/0.71
POL(LIST(x1)) = [3]x1    0.00/0.71
POL(c(x1)) = x1   
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

list(Cons(z0, z1)) → list(z1) 0.00/0.71
list(Nil) → True 0.00/0.71
list(Nil) → isEmpty[Match](Nil) 0.00/0.71
notEmpty(Cons(z0, z1)) → True 0.00/0.71
notEmpty(Nil) → False 0.00/0.71
goal(z0) → list(z0)
Tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))
S tuples:none
K tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))
Defined Rule Symbols:

list, notEmpty, goal

Defined Pair Symbols:

LIST

Compound Symbols:

c

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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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0.00/0.73 EOF