YES(O(1), O(n^1)) 0.00/0.71 YES(O(1), O(n^1)) 0.00/0.72 0.00/0.72 0.00/0.72 0.00/0.72 0.00/0.72 0.00/0.72 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.72 0.00/0.72 0.00/0.72
0.00/0.72 0.00/0.72 0.00/0.72
0.00/0.72
0.00/0.72

(0) Obligation:

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

f(f(a)) → f(g(n__f(n__a))) 0.00/0.72
f(X) → n__f(X) 0.00/0.72
an__a 0.00/0.72
activate(n__f(X)) → f(activate(X)) 0.00/0.72
activate(n__a) → a 0.00/0.72
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)) → f(g(n__f(n__a))) 0.00/0.72
f(z0) → n__f(z0) 0.00/0.72
an__a 0.00/0.72
activate(n__f(z0)) → f(activate(z0)) 0.00/0.72
activate(n__a) → a 0.00/0.72
activate(z0) → z0
Tuples:

F(f(a)) → c(F(g(n__f(n__a)))) 0.00/0.72
ACTIVATE(n__f(z0)) → c3(F(activate(z0)), ACTIVATE(z0)) 0.00/0.72
ACTIVATE(n__a) → c4(A)
S tuples:

F(f(a)) → c(F(g(n__f(n__a)))) 0.00/0.72
ACTIVATE(n__f(z0)) → c3(F(activate(z0)), ACTIVATE(z0)) 0.00/0.72
ACTIVATE(n__a) → c4(A)
K tuples:none
Defined Rule Symbols:

f, a, activate

Defined Pair Symbols:

F, ACTIVATE

Compound Symbols:

c, c3, c4

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(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

F(f(a)) → c(F(g(n__f(n__a))))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a)) → f(g(n__f(n__a))) 0.00/0.72
f(z0) → n__f(z0) 0.00/0.72
an__a 0.00/0.72
activate(n__f(z0)) → f(activate(z0)) 0.00/0.72
activate(n__a) → a 0.00/0.72
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c3(F(activate(z0)), ACTIVATE(z0)) 0.00/0.72
ACTIVATE(n__a) → c4(A)
S tuples:

ACTIVATE(n__f(z0)) → c3(F(activate(z0)), ACTIVATE(z0)) 0.00/0.72
ACTIVATE(n__a) → c4(A)
K tuples:none
Defined Rule Symbols:

f, a, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3, c4

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0.00/0.72

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

Removed 2 trailing tuple parts
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a)) → f(g(n__f(n__a))) 0.00/0.72
f(z0) → n__f(z0) 0.00/0.72
an__a 0.00/0.72
activate(n__f(z0)) → f(activate(z0)) 0.00/0.72
activate(n__a) → a 0.00/0.72
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c3(ACTIVATE(z0)) 0.00/0.72
ACTIVATE(n__a) → c4
S tuples:

ACTIVATE(n__f(z0)) → c3(ACTIVATE(z0)) 0.00/0.72
ACTIVATE(n__a) → c4
K tuples:none
Defined Rule Symbols:

f, a, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3, c4

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

Removed 1 trailing nodes:

ACTIVATE(n__a) → c4
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a)) → f(g(n__f(n__a))) 0.00/0.72
f(z0) → n__f(z0) 0.00/0.72
an__a 0.00/0.72
activate(n__f(z0)) → f(activate(z0)) 0.00/0.72
activate(n__a) → a 0.00/0.72
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c3(ACTIVATE(z0)) 0.00/0.72
ACTIVATE(n__a) → c4
S tuples:

ACTIVATE(n__f(z0)) → c3(ACTIVATE(z0)) 0.00/0.72
ACTIVATE(n__a) → c4
K tuples:none
Defined Rule Symbols:

f, a, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3, c4

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(9) 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)) → c3(ACTIVATE(z0)) 0.00/0.72
ACTIVATE(n__a) → c4
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__f(z0)) → c3(ACTIVATE(z0)) 0.00/0.72
ACTIVATE(n__a) → c4
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.72

POL(ACTIVATE(x1)) = [3] + [3]x1    0.00/0.72
POL(c3(x1)) = x1    0.00/0.72
POL(c4) = 0    0.00/0.72
POL(n__a) = [5]    0.00/0.72
POL(n__f(x1)) = [3] + x1   
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(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a)) → f(g(n__f(n__a))) 0.00/0.72
f(z0) → n__f(z0) 0.00/0.72
an__a 0.00/0.72
activate(n__f(z0)) → f(activate(z0)) 0.00/0.72
activate(n__a) → a 0.00/0.72
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c3(ACTIVATE(z0)) 0.00/0.72
ACTIVATE(n__a) → c4
S tuples:none
K tuples:

ACTIVATE(n__f(z0)) → c3(ACTIVATE(z0)) 0.00/0.72
ACTIVATE(n__a) → c4
Defined Rule Symbols:

f, a, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3, c4

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(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty
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(12) BOUNDS(O(1), O(1))

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0.00/0.78 EOF