YES(O(1), O(n^1)) 0.00/0.74 YES(O(1), O(n^1)) 0.00/0.76 0.00/0.76 0.00/0.76 0.00/0.76 0.00/0.76 0.00/0.76 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.76 0.00/0.76 0.00/0.76
0.00/0.76 0.00/0.76 0.00/0.76
0.00/0.76
0.00/0.76

(0) Obligation:

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

f(n__f(n__a)) → f(n__g(n__f(n__a))) 0.00/0.76
f(X) → n__f(X) 0.00/0.76
an__a 0.00/0.76
g(X) → n__g(X) 0.00/0.76
activate(n__f(X)) → f(X) 0.00/0.76
activate(n__a) → a 0.00/0.76
activate(n__g(X)) → g(activate(X)) 0.00/0.76
activate(X) → X

Rewrite Strategy: INNERMOST
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0.00/0.76

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
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0.00/0.76

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(n__f(n__a)) → f(n__g(n__f(n__a))) 0.00/0.76
f(z0) → n__f(z0) 0.00/0.76
an__a 0.00/0.76
g(z0) → n__g(z0) 0.00/0.76
activate(n__f(z0)) → f(z0) 0.00/0.76
activate(n__a) → a 0.00/0.76
activate(n__g(z0)) → g(activate(z0)) 0.00/0.76
activate(z0) → z0
Tuples:

F(n__f(n__a)) → c(F(n__g(n__f(n__a)))) 0.00/0.76
ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.76
ACTIVATE(n__a) → c5(A) 0.00/0.76
ACTIVATE(n__g(z0)) → c6(G(activate(z0)), ACTIVATE(z0))
S tuples:

F(n__f(n__a)) → c(F(n__g(n__f(n__a)))) 0.00/0.76
ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.76
ACTIVATE(n__a) → c5(A) 0.00/0.76
ACTIVATE(n__g(z0)) → c6(G(activate(z0)), ACTIVATE(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 3 trailing tuple parts
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0.00/0.76

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(n__f(n__a)) → f(n__g(n__f(n__a))) 0.00/0.76
f(z0) → n__f(z0) 0.00/0.76
an__a 0.00/0.76
g(z0) → n__g(z0) 0.00/0.76
activate(n__f(z0)) → f(z0) 0.00/0.76
activate(n__a) → a 0.00/0.76
activate(n__g(z0)) → g(activate(z0)) 0.00/0.76
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.76
F(n__f(n__a)) → c 0.00/0.76
ACTIVATE(n__a) → c5 0.00/0.76
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
S tuples:

ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.76
F(n__f(n__a)) → c 0.00/0.76
ACTIVATE(n__a) → c5 0.00/0.76
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

f, a, g, activate

Defined Pair Symbols:

ACTIVATE, F

Compound Symbols:

c4, c, c5, c6

0.00/0.76
0.00/0.76

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

Removed 3 trailing nodes:

ACTIVATE(n__a) → c5 0.00/0.76
ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.76
F(n__f(n__a)) → c
0.00/0.76
0.00/0.76

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(n__f(n__a)) → f(n__g(n__f(n__a))) 0.00/0.76
f(z0) → n__f(z0) 0.00/0.76
an__a 0.00/0.76
g(z0) → n__g(z0) 0.00/0.76
activate(n__f(z0)) → f(z0) 0.00/0.76
activate(n__a) → a 0.00/0.76
activate(n__g(z0)) → g(activate(z0)) 0.00/0.76
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.76
F(n__f(n__a)) → c 0.00/0.76
ACTIVATE(n__a) → c5 0.00/0.76
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
S tuples:

ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.76
F(n__f(n__a)) → c 0.00/0.76
ACTIVATE(n__a) → c5 0.00/0.76
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
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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0.00/0.76

(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__a) → c5 0.00/0.76
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.76
F(n__f(n__a)) → c 0.00/0.76
ACTIVATE(n__a) → c5 0.00/0.76
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.76

POL(ACTIVATE(x1)) = x1    0.00/0.76
POL(F(x1)) = 0    0.00/0.76
POL(c) = 0    0.00/0.76
POL(c4(x1)) = x1    0.00/0.76
POL(c5) = 0    0.00/0.76
POL(c6(x1)) = x1    0.00/0.76
POL(n__a) = [1]    0.00/0.76
POL(n__f(x1)) = 0    0.00/0.76
POL(n__g(x1)) = [1] + x1   
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(n__f(n__a)) → f(n__g(n__f(n__a))) 0.00/0.76
f(z0) → n__f(z0) 0.00/0.76
an__a 0.00/0.76
g(z0) → n__g(z0) 0.00/0.76
activate(n__f(z0)) → f(z0) 0.00/0.76
activate(n__a) → a 0.00/0.76
activate(n__g(z0)) → g(activate(z0)) 0.00/0.76
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.76
F(n__f(n__a)) → c 0.00/0.76
ACTIVATE(n__a) → c5 0.00/0.76
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
S tuples:

ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.76
F(n__f(n__a)) → c
K tuples:

ACTIVATE(n__a) → c5 0.00/0.76
ACTIVATE(n__g(z0)) → c6(ACTIVATE(z0))
Defined Rule Symbols:

f, a, g, activate

Defined Pair Symbols:

ACTIVATE, F

Compound Symbols:

c4, c, c5, c6

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0.00/0.76

(9) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

ACTIVATE(n__f(z0)) → c4(F(z0)) 0.00/0.76
F(n__f(n__a)) → c 0.00/0.76
F(n__f(n__a)) → c
Now S is empty
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(10) BOUNDS(O(1), O(1))

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0.00/0.78 EOF