YES(O(1), O(n^1)) 0.00/0.81 YES(O(1), O(n^1)) 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.84 0.00/0.84 0.00/0.84
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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))
0.00/0.84 
↳2 CdtProblem
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↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
0.00/0.84 
↳4 CdtProblem
0.00/0.84 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.84 
↳6 CdtProblem
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↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳8 CdtProblem
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↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳10 BOUNDS(O(1), O(1))
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0.00/0.84 0.00/0.84
0.00/0.84
0.00/0.84

(0) Obligation:

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

f(a) → b 0.00/0.84
f(c) → d 0.00/0.84
f(g(x, y)) → g(f(x), f(y)) 0.00/0.84
f(h(x, y)) → g(h(y, f(x)), h(x, f(y))) 0.00/0.84
g(x, x) → h(e, x)

Rewrite Strategy: INNERMOST
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0.00/0.84

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

Converted CpxTRS to CDT
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0.00/0.84

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a) → b 0.00/0.84
f(c) → d 0.00/0.84
f(g(z0, z1)) → g(f(z0), f(z1)) 0.00/0.84
f(h(z0, z1)) → g(h(z1, f(z0)), h(z0, f(z1))) 0.00/0.84
g(z0, z0) → h(e, z0)
Tuples:

F(g(z0, z1)) → c3(G(f(z0), f(z1)), F(z0), F(z1)) 0.00/0.84
F(h(z0, z1)) → c4(G(h(z1, f(z0)), h(z0, f(z1))), F(z0), F(z1))
S tuples:

F(g(z0, z1)) → c3(G(f(z0), f(z1)), F(z0), F(z1)) 0.00/0.84
F(h(z0, z1)) → c4(G(h(z1, f(z0)), h(z0, f(z1))), F(z0), F(z1))
K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

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0.00/0.84

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

Removed 2 trailing tuple parts
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0.00/0.84

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a) → b 0.00/0.84
f(c) → d 0.00/0.84
f(g(z0, z1)) → g(f(z0), f(z1)) 0.00/0.84
f(h(z0, z1)) → g(h(z1, f(z0)), h(z0, f(z1))) 0.00/0.84
g(z0, z0) → h(e, z0)
Tuples:

F(g(z0, z1)) → c3(F(z0), F(z1)) 0.00/0.84
F(h(z0, z1)) → c4(F(z0), F(z1))
S tuples:

F(g(z0, z1)) → c3(F(z0), F(z1)) 0.00/0.84
F(h(z0, z1)) → c4(F(z0), F(z1))
K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

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0.00/0.84

(5) 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.

F(g(z0, z1)) → c3(F(z0), F(z1))
We considered the (Usable) Rules:none
And the Tuples:

F(g(z0, z1)) → c3(F(z0), F(z1)) 0.00/0.84
F(h(z0, z1)) → c4(F(z0), F(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.84

POL(F(x1)) = [2]x1    0.00/0.84
POL(c3(x1, x2)) = x1 + x2    0.00/0.84
POL(c4(x1, x2)) = x1 + x2    0.00/0.84
POL(g(x1, x2)) = [3] + [4]x1 + [2]x2    0.00/0.84
POL(h(x1, x2)) = x1 + x2   
0.00/0.84
0.00/0.84

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a) → b 0.00/0.84
f(c) → d 0.00/0.84
f(g(z0, z1)) → g(f(z0), f(z1)) 0.00/0.84
f(h(z0, z1)) → g(h(z1, f(z0)), h(z0, f(z1))) 0.00/0.84
g(z0, z0) → h(e, z0)
Tuples:

F(g(z0, z1)) → c3(F(z0), F(z1)) 0.00/0.84
F(h(z0, z1)) → c4(F(z0), F(z1))
S tuples:

F(h(z0, z1)) → c4(F(z0), F(z1))
K tuples:

F(g(z0, z1)) → c3(F(z0), F(z1))
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

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0.00/0.84

(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.

F(h(z0, z1)) → c4(F(z0), F(z1))
We considered the (Usable) Rules:none
And the Tuples:

F(g(z0, z1)) → c3(F(z0), F(z1)) 0.00/0.84
F(h(z0, z1)) → c4(F(z0), F(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.84

POL(F(x1)) = x1    0.00/0.84
POL(c3(x1, x2)) = x1 + x2    0.00/0.84
POL(c4(x1, x2)) = x1 + x2    0.00/0.84
POL(g(x1, x2)) = [4]x1 + [4]x2    0.00/0.84
POL(h(x1, x2)) = [1] + x1 + x2   
0.00/0.84
0.00/0.84

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a) → b 0.00/0.84
f(c) → d 0.00/0.84
f(g(z0, z1)) → g(f(z0), f(z1)) 0.00/0.84
f(h(z0, z1)) → g(h(z1, f(z0)), h(z0, f(z1))) 0.00/0.84
g(z0, z0) → h(e, z0)
Tuples:

F(g(z0, z1)) → c3(F(z0), F(z1)) 0.00/0.84
F(h(z0, z1)) → c4(F(z0), F(z1))
S tuples:none
K tuples:

F(g(z0, z1)) → c3(F(z0), F(z1)) 0.00/0.84
F(h(z0, z1)) → c4(F(z0), F(z1))
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

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0.00/0.84

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

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

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0.00/0.86 EOF