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:

g(X) → u(h(X), h(X), X) 0.00/0.56
u(d, c(Y), X) → k(Y) 0.00/0.56
h(d) → c(a) 0.00/0.56
h(d) → c(b) 0.00/0.56
f(k(a), k(b), X) → f(X, 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:

g(z0) → u(h(z0), h(z0), z0) 0.00/0.56
u(d, c(z0), z1) → k(z0) 0.00/0.56
h(d) → c(a) 0.00/0.56
h(d) → c(b) 0.00/0.56
f(k(a), k(b), z0) → f(z0, z0, z0)
Tuples:

G(z0) → c1(U(h(z0), h(z0), z0), H(z0), H(z0)) 0.00/0.56
F(k(a), k(b), z0) → c5(F(z0, z0, z0))
S tuples:

G(z0) → c1(U(h(z0), h(z0), z0), H(z0), H(z0)) 0.00/0.56
F(k(a), k(b), z0) → c5(F(z0, z0, z0))
K tuples:none
Defined Rule Symbols:

g, u, h, f

Defined Pair Symbols:

G, F

Compound Symbols:

c1, c5

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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:

g(z0) → u(h(z0), h(z0), z0) 0.00/0.56
u(d, c(z0), z1) → k(z0) 0.00/0.56
h(d) → c(a) 0.00/0.56
h(d) → c(b) 0.00/0.56
f(k(a), k(b), z0) → f(z0, z0, z0)
Tuples:

G(z0) → c1 0.00/0.56
F(k(a), k(b), z0) → c5
S tuples:

G(z0) → c1 0.00/0.56
F(k(a), k(b), z0) → c5
K tuples:none
Defined Rule Symbols:

g, u, h, f

Defined Pair Symbols:

G, F

Compound Symbols:

c1, c5

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

Removed 2 trailing nodes:

G(z0) → c1 0.00/0.56
F(k(a), k(b), z0) → c5
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(z0) → u(h(z0), h(z0), z0) 0.00/0.56
u(d, c(z0), z1) → k(z0) 0.00/0.56
h(d) → c(a) 0.00/0.56
h(d) → c(b) 0.00/0.56
f(k(a), k(b), z0) → f(z0, z0, z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

g, u, h, f

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