YES(O(1), O(n^1)) 19.63/6.75 YES(O(1), O(n^1)) 19.63/6.76 19.63/6.76 19.63/6.76 19.63/6.76 19.63/6.76 19.63/6.76 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 19.63/6.76 19.63/6.76 19.63/6.76
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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 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
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↳4 CdtProblem
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↳5 CdtUnreachableProof (⇔)
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↳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) Obligation:

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

h(z, e(x)) → h(c(z), d(z, x)) 19.63/6.76
d(z, g(0, 0)) → e(0) 19.63/6.76
d(z, g(x, y)) → g(e(x), d(z, y)) 19.63/6.76
d(c(z), g(g(x, y), 0)) → g(d(c(z), g(x, y)), d(z, g(x, y))) 19.63/6.76
g(e(x), e(y)) → e(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:

h(z0, e(z1)) → h(c(z0), d(z0, z1)) 19.63/6.76
d(z0, g(0, 0)) → e(0) 19.63/6.76
d(z0, g(z1, z2)) → g(e(z1), d(z0, z2)) 19.63/6.76
d(c(z0), g(g(z1, z2), 0)) → g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2))) 19.63/6.76
g(e(z0), e(z1)) → e(g(z0, z1))
Tuples:

H(z0, e(z1)) → c1(H(c(z0), d(z0, z1)), D(z0, z1)) 19.63/6.76
D(z0, g(z1, z2)) → c3(G(e(z1), d(z0, z2)), D(z0, z2)) 19.63/6.76
D(c(z0), g(g(z1, z2), 0)) → c4(G(d(c(z0), g(z1, z2)), d(z0, g(z1, z2))), D(c(z0), g(z1, z2)), G(z1, z2), D(z0, g(z1, z2)), G(z1, z2)) 19.63/6.76
G(e(z0), e(z1)) → c5(G(z0, z1))
S tuples:

H(z0, e(z1)) → c1(H(c(z0), d(z0, z1)), D(z0, z1)) 19.63/6.76
D(z0, g(z1, z2)) → c3(G(e(z1), d(z0, z2)), D(z0, z2)) 19.63/6.76
D(c(z0), g(g(z1, z2), 0)) → c4(G(d(c(z0), g(z1, z2)), d(z0, g(z1, z2))), D(c(z0), g(z1, z2)), G(z1, z2), D(z0, g(z1, z2)), G(z1, z2)) 19.63/6.76
G(e(z0), e(z1)) → c5(G(z0, z1))
K tuples:none
Defined Rule Symbols:

h, d, g

Defined Pair Symbols:

H, D, G

Compound Symbols:

c1, c3, c4, c5

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

Use narrowing to replace H(z0, e(z1)) → c1(H(c(z0), d(z0, z1)), D(z0, z1)) by

H(z0, e(g(0, 0))) → c1(H(c(z0), e(0)), D(z0, g(0, 0))) 19.63/6.76
H(z0, e(g(z1, z2))) → c1(H(c(z0), g(e(z1), d(z0, z2))), D(z0, g(z1, z2))) 19.63/6.76
H(c(z0), e(g(g(z1, z2), 0))) → c1(H(c(c(z0)), g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2)))), D(c(z0), g(g(z1, z2), 0)))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(z0, e(z1)) → h(c(z0), d(z0, z1)) 19.63/6.76
d(z0, g(0, 0)) → e(0) 19.63/6.76
d(z0, g(z1, z2)) → g(e(z1), d(z0, z2)) 19.63/6.76
d(c(z0), g(g(z1, z2), 0)) → g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2))) 19.63/6.76
g(e(z0), e(z1)) → e(g(z0, z1))
Tuples:

D(z0, g(z1, z2)) → c3(G(e(z1), d(z0, z2)), D(z0, z2)) 19.63/6.76
D(c(z0), g(g(z1, z2), 0)) → c4(G(d(c(z0), g(z1, z2)), d(z0, g(z1, z2))), D(c(z0), g(z1, z2)), G(z1, z2), D(z0, g(z1, z2)), G(z1, z2)) 19.63/6.76
G(e(z0), e(z1)) → c5(G(z0, z1)) 19.63/6.76
H(z0, e(g(0, 0))) → c1(H(c(z0), e(0)), D(z0, g(0, 0))) 19.63/6.76
H(z0, e(g(z1, z2))) → c1(H(c(z0), g(e(z1), d(z0, z2))), D(z0, g(z1, z2))) 19.63/6.76
H(c(z0), e(g(g(z1, z2), 0))) → c1(H(c(c(z0)), g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2)))), D(c(z0), g(g(z1, z2), 0)))
S tuples:

D(z0, g(z1, z2)) → c3(G(e(z1), d(z0, z2)), D(z0, z2)) 19.63/6.76
D(c(z0), g(g(z1, z2), 0)) → c4(G(d(c(z0), g(z1, z2)), d(z0, g(z1, z2))), D(c(z0), g(z1, z2)), G(z1, z2), D(z0, g(z1, z2)), G(z1, z2)) 19.63/6.76
G(e(z0), e(z1)) → c5(G(z0, z1)) 19.63/6.76
H(z0, e(g(0, 0))) → c1(H(c(z0), e(0)), D(z0, g(0, 0))) 19.63/6.77
H(z0, e(g(z1, z2))) → c1(H(c(z0), g(e(z1), d(z0, z2))), D(z0, g(z1, z2))) 19.63/6.77
H(c(z0), e(g(g(z1, z2), 0))) → c1(H(c(c(z0)), g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2)))), D(c(z0), g(g(z1, z2), 0)))
K tuples:none
Defined Rule Symbols:

h, d, g

Defined Pair Symbols:

D, G, H

Compound Symbols:

c3, c4, c5, c1

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(5) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

D(z0, g(z1, z2)) → c3(G(e(z1), d(z0, z2)), D(z0, z2)) 19.63/6.77
D(c(z0), g(g(z1, z2), 0)) → c4(G(d(c(z0), g(z1, z2)), d(z0, g(z1, z2))), D(c(z0), g(z1, z2)), G(z1, z2), D(z0, g(z1, z2)), G(z1, z2)) 19.63/6.77
H(z0, e(g(0, 0))) → c1(H(c(z0), e(0)), D(z0, g(0, 0))) 19.63/6.77
H(z0, e(g(z1, z2))) → c1(H(c(z0), g(e(z1), d(z0, z2))), D(z0, g(z1, z2))) 19.63/6.77
H(c(z0), e(g(g(z1, z2), 0))) → c1(H(c(c(z0)), g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2)))), D(c(z0), g(g(z1, z2), 0)))
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(z0, e(z1)) → h(c(z0), d(z0, z1)) 19.63/6.77
d(z0, g(0, 0)) → e(0) 19.63/6.77
d(z0, g(z1, z2)) → g(e(z1), d(z0, z2)) 19.63/6.77
d(c(z0), g(g(z1, z2), 0)) → g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2))) 19.63/6.77
g(e(z0), e(z1)) → e(g(z0, z1))
Tuples:

G(e(z0), e(z1)) → c5(G(z0, z1))
S tuples:

G(e(z0), e(z1)) → c5(G(z0, z1))
K tuples:none
Defined Rule Symbols:

h, d, g

Defined Pair Symbols:

G

Compound Symbols:

c5

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(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.

G(e(z0), e(z1)) → c5(G(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

G(e(z0), e(z1)) → c5(G(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 19.63/6.77

POL(G(x1, x2)) = [2]x2    19.63/6.77
POL(c5(x1)) = x1    19.63/6.77
POL(e(x1)) = [1] + x1   
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(z0, e(z1)) → h(c(z0), d(z0, z1)) 19.63/6.77
d(z0, g(0, 0)) → e(0) 19.63/6.77
d(z0, g(z1, z2)) → g(e(z1), d(z0, z2)) 19.63/6.77
d(c(z0), g(g(z1, z2), 0)) → g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2))) 19.63/6.77
g(e(z0), e(z1)) → e(g(z0, z1))
Tuples:

G(e(z0), e(z1)) → c5(G(z0, z1))
S tuples:none
K tuples:

G(e(z0), e(z1)) → c5(G(z0, z1))
Defined Rule Symbols:

h, d, g

Defined Pair Symbols:

G

Compound Symbols:

c5

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(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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19.93/6.83 EOF