YES(O(1), O(n^1)) 0.00/0.72 YES(O(1), O(n^1)) 0.00/0.73 0.00/0.73 0.00/0.73 0.00/0.73 0.00/0.73 0.00/0.73 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.73 0.00/0.73 0.00/0.73
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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:

a__g(X) → a__h(X) 0.00/0.73
a__c → d 0.00/0.73
a__h(d) → a__g(c) 0.00/0.73
mark(g(X)) → a__g(X) 0.00/0.73
mark(h(X)) → a__h(X) 0.00/0.73
mark(c) → a__c 0.00/0.73
mark(d) → d 0.00/0.73
a__g(X) → g(X) 0.00/0.73
a__h(X) → h(X) 0.00/0.73
a__c → c

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:

a__g(z0) → a__h(z0) 0.00/0.73
a__g(z0) → g(z0) 0.00/0.73
a__c → d 0.00/0.73
a__c → c 0.00/0.73
a__h(d) → a__g(c) 0.00/0.73
a__h(z0) → h(z0) 0.00/0.73
mark(g(z0)) → a__g(z0) 0.00/0.73
mark(h(z0)) → a__h(z0) 0.00/0.73
mark(c) → a__c 0.00/0.73
mark(d) → d
Tuples:

A__G(z0) → c1(A__H(z0)) 0.00/0.73
A__H(d) → c5(A__G(c)) 0.00/0.73
MARK(g(z0)) → c7(A__G(z0)) 0.00/0.73
MARK(h(z0)) → c8(A__H(z0)) 0.00/0.73
MARK(c) → c9(A__C)
S tuples:

A__G(z0) → c1(A__H(z0)) 0.00/0.73
A__H(d) → c5(A__G(c)) 0.00/0.73
MARK(g(z0)) → c7(A__G(z0)) 0.00/0.73
MARK(h(z0)) → c8(A__H(z0)) 0.00/0.73
MARK(c) → c9(A__C)
K tuples:none
Defined Rule Symbols:

a__g, a__c, a__h, mark

Defined Pair Symbols:

A__G, A__H, MARK

Compound Symbols:

c1, c5, c7, c8, c9

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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:

a__g(z0) → a__h(z0) 0.00/0.73
a__g(z0) → g(z0) 0.00/0.73
a__c → d 0.00/0.73
a__c → c 0.00/0.73
a__h(d) → a__g(c) 0.00/0.73
a__h(z0) → h(z0) 0.00/0.73
mark(g(z0)) → a__g(z0) 0.00/0.73
mark(h(z0)) → a__h(z0) 0.00/0.73
mark(c) → a__c 0.00/0.73
mark(d) → d
Tuples:

A__G(z0) → c1(A__H(z0)) 0.00/0.73
A__H(d) → c5(A__G(c)) 0.00/0.73
MARK(g(z0)) → c7(A__G(z0)) 0.00/0.73
MARK(h(z0)) → c8(A__H(z0)) 0.00/0.73
MARK(c) → c9
S tuples:

A__G(z0) → c1(A__H(z0)) 0.00/0.73
A__H(d) → c5(A__G(c)) 0.00/0.73
MARK(g(z0)) → c7(A__G(z0)) 0.00/0.73
MARK(h(z0)) → c8(A__H(z0)) 0.00/0.73
MARK(c) → c9
K tuples:none
Defined Rule Symbols:

a__g, a__c, a__h, mark

Defined Pair Symbols:

A__G, A__H, MARK

Compound Symbols:

c1, c5, c7, c8, c9

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(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

MARK(h(z0)) → c8(A__H(z0)) 0.00/0.73
MARK(g(z0)) → c7(A__G(z0))
Removed 1 trailing nodes:

MARK(c) → c9
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__g(z0) → a__h(z0) 0.00/0.73
a__g(z0) → g(z0) 0.00/0.73
a__c → d 0.00/0.73
a__c → c 0.00/0.73
a__h(d) → a__g(c) 0.00/0.73
a__h(z0) → h(z0) 0.00/0.73
mark(g(z0)) → a__g(z0) 0.00/0.73
mark(h(z0)) → a__h(z0) 0.00/0.73
mark(c) → a__c 0.00/0.73
mark(d) → d
Tuples:

A__G(z0) → c1(A__H(z0)) 0.00/0.73
A__H(d) → c5(A__G(c))
S tuples:

A__G(z0) → c1(A__H(z0)) 0.00/0.73
A__H(d) → c5(A__G(c))
K tuples:none
Defined Rule Symbols:

a__g, a__c, a__h, mark

Defined Pair Symbols:

A__G, A__H

Compound Symbols:

c1, c5

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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.

A__G(z0) → c1(A__H(z0)) 0.00/0.73
A__H(d) → c5(A__G(c))
We considered the (Usable) Rules:none
And the Tuples:

A__G(z0) → c1(A__H(z0)) 0.00/0.73
A__H(d) → c5(A__G(c))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.73

POL(A__G(x1)) = [4] + [4]x1    0.00/0.73
POL(A__H(x1)) = [2] + [4]x1    0.00/0.73
POL(c) = [2]    0.00/0.73
POL(c1(x1)) = x1    0.00/0.73
POL(c5(x1)) = x1    0.00/0.73
POL(d) = [4]   
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__g(z0) → a__h(z0) 0.00/0.73
a__g(z0) → g(z0) 0.00/0.73
a__c → d 0.00/0.73
a__c → c 0.00/0.73
a__h(d) → a__g(c) 0.00/0.73
a__h(z0) → h(z0) 0.00/0.73
mark(g(z0)) → a__g(z0) 0.00/0.73
mark(h(z0)) → a__h(z0) 0.00/0.73
mark(c) → a__c 0.00/0.73
mark(d) → d
Tuples:

A__G(z0) → c1(A__H(z0)) 0.00/0.73
A__H(d) → c5(A__G(c))
S tuples:none
K tuples:

A__G(z0) → c1(A__H(z0)) 0.00/0.73
A__H(d) → c5(A__G(c))
Defined Rule Symbols:

a__g, a__c, a__h, mark

Defined Pair Symbols:

A__G, A__H

Compound Symbols:

c1, c5

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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