YES(O(1), O(1)) 0.00/0.54 YES(O(1), O(1)) 0.00/0.55 0.00/0.55 0.00/0.55 0.00/0.55 0.00/0.55 0.00/0.55 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.55 0.00/0.55 0.00/0.55
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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:

p(f(f(x))) → q(f(g(x))) 0.00/0.55
p(g(g(x))) → q(g(f(x))) 0.00/0.55
q(f(f(x))) → p(f(g(x))) 0.00/0.55
q(g(g(x))) → p(g(f(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:

p(f(f(z0))) → q(f(g(z0))) 0.00/0.55
p(g(g(z0))) → q(g(f(z0))) 0.00/0.55
q(f(f(z0))) → p(f(g(z0))) 0.00/0.55
q(g(g(z0))) → p(g(f(z0)))
Tuples:

P(f(f(z0))) → c(Q(f(g(z0)))) 0.00/0.55
P(g(g(z0))) → c1(Q(g(f(z0)))) 0.00/0.55
Q(f(f(z0))) → c2(P(f(g(z0)))) 0.00/0.55
Q(g(g(z0))) → c3(P(g(f(z0))))
S tuples:

P(f(f(z0))) → c(Q(f(g(z0)))) 0.00/0.55
P(g(g(z0))) → c1(Q(g(f(z0)))) 0.00/0.55
Q(f(f(z0))) → c2(P(f(g(z0)))) 0.00/0.55
Q(g(g(z0))) → c3(P(g(f(z0))))
K tuples:none
Defined Rule Symbols:

p, q

Defined Pair Symbols:

P, Q

Compound Symbols:

c, c1, c2, c3

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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:

p(f(f(z0))) → q(f(g(z0))) 0.00/0.55
p(g(g(z0))) → q(g(f(z0))) 0.00/0.55
q(f(f(z0))) → p(f(g(z0))) 0.00/0.55
q(g(g(z0))) → p(g(f(z0)))
Tuples:

P(f(f(z0))) → c 0.00/0.55
P(g(g(z0))) → c1 0.00/0.55
Q(f(f(z0))) → c2 0.00/0.55
Q(g(g(z0))) → c3
S tuples:

P(f(f(z0))) → c 0.00/0.55
P(g(g(z0))) → c1 0.00/0.55
Q(f(f(z0))) → c2 0.00/0.55
Q(g(g(z0))) → c3
K tuples:none
Defined Rule Symbols:

p, q

Defined Pair Symbols:

P, Q

Compound Symbols:

c, c1, c2, c3

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

Removed 4 trailing nodes:

P(g(g(z0))) → c1 0.00/0.55
Q(g(g(z0))) → c3 0.00/0.55
Q(f(f(z0))) → c2 0.00/0.55
P(f(f(z0))) → c
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(f(f(z0))) → q(f(g(z0))) 0.00/0.55
p(g(g(z0))) → q(g(f(z0))) 0.00/0.55
q(f(f(z0))) → p(f(g(z0))) 0.00/0.55
q(g(g(z0))) → p(g(f(z0)))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

p, q

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