YES(O(1), O(n^1)) 0.00/0.73 YES(O(1), O(n^1)) 0.00/0.75 0.00/0.75 0.00/0.75 0.00/0.75 0.00/0.75 0.00/0.75 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.75 0.00/0.75 0.00/0.75
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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 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳4 CdtProblem
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↳5 CdtKnowledgeProof (⇔)
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↳6 BOUNDS(O(1), O(1))
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(0) Obligation:

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

f(x, y) → g(x, y) 0.00/0.75
g(h(x), y) → h(f(x, y)) 0.00/0.75
g(h(x), y) → h(g(x, y))

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(z0, z1) → g(z0, z1) 0.00/0.75
g(h(z0), z1) → h(f(z0, z1)) 0.00/0.75
g(h(z0), z1) → h(g(z0, z1))
Tuples:

F(z0, z1) → c(G(z0, z1)) 0.00/0.75
G(h(z0), z1) → c1(F(z0, z1)) 0.00/0.75
G(h(z0), z1) → c2(G(z0, z1))
S tuples:

F(z0, z1) → c(G(z0, z1)) 0.00/0.75
G(h(z0), z1) → c1(F(z0, z1)) 0.00/0.75
G(h(z0), z1) → c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1, c2

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(3) 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.

G(h(z0), z1) → c1(F(z0, z1)) 0.00/0.75
G(h(z0), z1) → c2(G(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

F(z0, z1) → c(G(z0, z1)) 0.00/0.75
G(h(z0), z1) → c1(F(z0, z1)) 0.00/0.75
G(h(z0), z1) → c2(G(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.75

POL(F(x1, x2)) = [2]x1    0.00/0.75
POL(G(x1, x2)) = [2]x1    0.00/0.75
POL(c(x1)) = x1    0.00/0.75
POL(c1(x1)) = x1    0.00/0.75
POL(c2(x1)) = x1    0.00/0.75
POL(h(x1)) = [3] + x1   
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z1) → g(z0, z1) 0.00/0.75
g(h(z0), z1) → h(f(z0, z1)) 0.00/0.75
g(h(z0), z1) → h(g(z0, z1))
Tuples:

F(z0, z1) → c(G(z0, z1)) 0.00/0.75
G(h(z0), z1) → c1(F(z0, z1)) 0.00/0.75
G(h(z0), z1) → c2(G(z0, z1))
S tuples:

F(z0, z1) → c(G(z0, z1))
K tuples:

G(h(z0), z1) → c1(F(z0, z1)) 0.00/0.75
G(h(z0), z1) → c2(G(z0, z1))
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1, c2

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(5) CdtKnowledgeProof (EQUIVALENT transformation)

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

F(z0, z1) → c(G(z0, z1)) 0.00/0.75
G(h(z0), z1) → c1(F(z0, z1)) 0.00/0.75
G(h(z0), z1) → c2(G(z0, z1))
Now S is empty
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(6) BOUNDS(O(1), O(1))

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0.00/0.78 EOF