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(n__f(n__a)) → f(n__g(f(n__a))) 0.00/0.56
f(X) → n__f(X) 0.00/0.56
a → n__a 0.00/0.56
g(X) → n__g(X) 0.00/0.56
activate(n__f(X)) → f(X) 0.00/0.56
activate(n__a) → a 0.00/0.56
activate(n__g(X)) → g(X) 0.00/0.56
activate(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:

f(n__f(n__a)) → f(n__g(f(n__a))) 0.00/0.56
f(z0) → n__f(z0) 0.00/0.56
a → n__a 0.00/0.56
g(z0) → n__g(z0) 0.00/0.56
activate(n__f(z0)) → f(z0) 0.00/0.56
activate(n__a) → a 0.00/0.56
activate(n__g(z0)) → g(z0) 0.00/0.56
activate(z0) → z0
Tuples:

F(n__f(n__a)) → c(F(n__g(f(n__a))), F(n__a)) 0.00/0.56
ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.56
ACTIVATE(n__a) → c5(A) 0.00/0.56
ACTIVATE(n__g(z0)) → c6(G(z0))
S tuples:

F(n__f(n__a)) → c(F(n__g(f(n__a))), F(n__a)) 0.00/0.56
ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.56
ACTIVATE(n__a) → c5(A) 0.00/0.56
ACTIVATE(n__g(z0)) → c6(G(z0))
K tuples:none
Defined Rule Symbols:

f, a, g, activate

Defined Pair Symbols:

F, ACTIVATE

Compound Symbols:

c, c4, c5, c6

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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(n__f(n__a)) → f(n__g(f(n__a))) 0.00/0.56
f(z0) → n__f(z0) 0.00/0.56
a → n__a 0.00/0.56
g(z0) → n__g(z0) 0.00/0.56
activate(n__f(z0)) → f(z0) 0.00/0.56
activate(n__a) → a 0.00/0.56
activate(n__g(z0)) → g(z0) 0.00/0.56
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.56
F(n__f(n__a)) → c 0.00/0.56
ACTIVATE(n__a) → c5 0.00/0.56
ACTIVATE(n__g(z0)) → c6
S tuples:

ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.56
F(n__f(n__a)) → c 0.00/0.56
ACTIVATE(n__a) → c5 0.00/0.56
ACTIVATE(n__g(z0)) → c6
K tuples:none
Defined Rule Symbols:

f, a, g, activate

Defined Pair Symbols:

ACTIVATE, F

Compound Symbols:

c4, c, c5, c6

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

Removed 4 trailing nodes:

F(n__f(n__a)) → c 0.00/0.56
ACTIVATE(n__g(z0)) → c6 0.00/0.56
ACTIVATE(n__a) → c5 0.00/0.56
ACTIVATE(n__f(z0)) → c4(F(z0))
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(n__f(n__a)) → f(n__g(f(n__a))) 0.00/0.56
f(z0) → n__f(z0) 0.00/0.56
a → n__a 0.00/0.56
g(z0) → n__g(z0) 0.00/0.56
activate(n__f(z0)) → f(z0) 0.00/0.56
activate(n__a) → a 0.00/0.56
activate(n__g(z0)) → g(z0) 0.00/0.56
activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, a, g, activate

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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0.00/0.58 EOF