YES(O(1), O(1)) 0.00/0.54 YES(O(1), O(1)) 0.00/0.56 0.00/0.56 0.00/0.56 0.00/0.56 0.00/0.56 0.00/0.56 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.56 0.00/0.56 0.00/0.56
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(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 (BOTH BOUNDS(ID, ID))
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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:

f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d')) 0.00/0.56
f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))

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:

f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d')) 0.00/0.56
f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))
Tuples:

F(g(i(a, b, b'), c), d) → c1(F(.(b, c), d'), F(.(b', c), d')) 0.00/0.56
F(g(h(a, b), c), d) → c2(F(.(b, g(h(a, b), c)), d), F(c, d'))
S tuples:

F(g(i(a, b, b'), c), d) → c1(F(.(b, c), d'), F(.(b', c), d')) 0.00/0.56
F(g(h(a, b), c), d) → c2(F(.(b, g(h(a, b), c)), d), F(c, d'))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c2

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

Removed 4 trailing tuple parts
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d')) 0.00/0.56
f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))
Tuples:

F(g(i(a, b, b'), c), d) → c1 0.00/0.56
F(g(h(a, b), c), d) → c2
S tuples:

F(g(i(a, b, b'), c), d) → c1 0.00/0.56
F(g(h(a, b), c), d) → c2
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c2

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

Removed 2 trailing nodes:

F(g(h(a, b), c), d) → c2 0.00/0.56
F(g(i(a, b, b'), c), d) → c1
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d')) 0.00/0.56
f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:none

Compound Symbols:none

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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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0.00/0.57 EOF