YES(O(1), O(n^1)) 0.00/0.73 YES(O(1), O(n^1)) 0.00/0.74 0.00/0.74 0.00/0.74 0.00/0.74 0.00/0.74 0.00/0.74 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.74 0.00/0.74 0.00/0.74
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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 (BOTH BOUNDS(ID, ID))
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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:

f(f(a)) → c(n__f(n__g(n__f(n__a)))) 0.00/0.74
f(X) → n__f(X) 0.00/0.74
g(X) → n__g(X) 0.00/0.74
a → n__a 0.00/0.74
activate(n__f(X)) → f(activate(X)) 0.00/0.74
activate(n__g(X)) → g(activate(X)) 0.00/0.74
activate(n__a) → a 0.00/0.74
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(f(a)) → c(n__f(n__g(n__f(n__a)))) 0.00/0.74
f(z0) → n__f(z0) 0.00/0.74
g(z0) → n__g(z0) 0.00/0.74
a → n__a 0.00/0.74
activate(n__f(z0)) → f(activate(z0)) 0.00/0.74
activate(n__g(z0)) → g(activate(z0)) 0.00/0.74
activate(n__a) → a 0.00/0.74
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c5(F(activate(z0)), ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__g(z0)) → c6(G(activate(z0)), ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__a) → c7(A)
S tuples:

ACTIVATE(n__f(z0)) → c5(F(activate(z0)), ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__g(z0)) → c6(G(activate(z0)), ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__a) → c7(A)
K tuples:none
Defined Rule Symbols:

f, g, a, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6, c7

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

Removed 3 trailing tuple parts
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a)) → c(n__f(n__g(n__f(n__a)))) 0.00/0.74
f(z0) → n__f(z0) 0.00/0.74
g(z0) → n__g(z0) 0.00/0.74
a → n__a 0.00/0.74
activate(n__f(z0)) → f(activate(z0)) 0.00/0.74
activate(n__g(z0)) → g(activate(z0)) 0.00/0.74
activate(n__a) → a 0.00/0.74
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__a) → c7
S tuples:

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__a) → c7
K tuples:none
Defined Rule Symbols:

f, g, a, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6, c7

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

Removed 1 trailing nodes:

ACTIVATE(n__a) → c7
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a)) → c(n__f(n__g(n__f(n__a)))) 0.00/0.74
f(z0) → n__f(z0) 0.00/0.74
g(z0) → n__g(z0) 0.00/0.74
a → n__a 0.00/0.74
activate(n__f(z0)) → f(activate(z0)) 0.00/0.74
activate(n__g(z0)) → g(activate(z0)) 0.00/0.74
activate(n__a) → a 0.00/0.74
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__a) → c7
S tuples:

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__a) → c7
K tuples:none
Defined Rule Symbols:

f, g, a, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6, c7

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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.

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__a) → c7
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__a) → c7
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.74

POL(ACTIVATE(x1)) = [5] + [3]x1    0.00/0.74
POL(c5(x1)) = x1    0.00/0.74
POL(c6(x1)) = x1    0.00/0.74
POL(c7) = 0    0.00/0.74
POL(n__a) = [5]    0.00/0.74
POL(n__f(x1)) = [5] + x1    0.00/0.74
POL(n__g(x1)) = [5] + x1   
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a)) → c(n__f(n__g(n__f(n__a)))) 0.00/0.74
f(z0) → n__f(z0) 0.00/0.74
g(z0) → n__g(z0) 0.00/0.74
a → n__a 0.00/0.74
activate(n__f(z0)) → f(activate(z0)) 0.00/0.74
activate(n__g(z0)) → g(activate(z0)) 0.00/0.74
activate(n__a) → a 0.00/0.74
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__a) → c7
S tuples:none
K tuples:

ACTIVATE(n__f(z0)) → c5(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__a) → c7
Defined Rule Symbols:

f, g, a, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6, c7

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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