YES(O(1), O(n^1)) 16.47/6.80 YES(O(1), O(n^1)) 16.79/6.87 16.79/6.87 16.79/6.87 16.79/6.87 16.79/6.87 16.79/6.87 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 16.79/6.87 16.79/6.87 16.79/6.87
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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
16.79/6.87 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
16.79/6.87 
↳2 CdtProblem
16.79/6.87 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
16.79/6.87 
↳4 CdtProblem
16.79/6.87 
↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
16.79/6.87 
↳6 CdtProblem
16.79/6.87 
↳7 CdtUnreachableProof (⇔)
16.79/6.87 
↳8 CdtProblem
16.79/6.87 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
16.79/6.87 
↳10 CdtProblem
16.79/6.87 
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID))
16.79/6.87 
↳12 BOUNDS(O(1), O(1))
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16.79/6.87

(0) Obligation:

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

h(e(x), y) → h(d(x, y), s(y)) 16.79/6.87
d(g(g(0, x), y), s(z)) → g(e(x), d(g(g(0, x), y), z)) 16.79/6.87
d(g(g(0, x), y), 0) → e(y) 16.79/6.87
d(g(0, x), y) → e(x) 16.79/6.87
d(g(x, y), z) → g(d(x, z), e(y)) 16.79/6.87
g(e(x), e(y)) → e(g(x, y))

Rewrite Strategy: INNERMOST
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16.79/6.87

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

Converted CpxTRS to CDT
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16.79/6.87

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(e(z0), z1) → h(d(z0, z1), s(z1)) 16.79/6.87
d(g(g(0, z0), z1), s(z2)) → g(e(z0), d(g(g(0, z0), z1), z2)) 16.79/6.87
d(g(g(0, z0), z1), 0) → e(z1) 16.79/6.87
d(g(0, z0), z1) → e(z0) 16.79/6.87
d(g(z0, z1), z2) → g(d(z0, z2), e(z1)) 16.79/6.87
g(e(z0), e(z1)) → e(g(z0, z1))
Tuples:

H(e(z0), z1) → c(H(d(z0, z1), s(z1)), D(z0, z1)) 16.79/6.87
D(g(g(0, z0), z1), s(z2)) → c1(G(e(z0), d(g(g(0, z0), z1), z2)), D(g(g(0, z0), z1), z2), G(g(0, z0), z1), G(0, z0)) 16.79/6.87
D(g(z0, z1), z2) → c4(G(d(z0, z2), e(z1)), D(z0, z2)) 16.79/6.87
G(e(z0), e(z1)) → c5(G(z0, z1))
S tuples:

H(e(z0), z1) → c(H(d(z0, z1), s(z1)), D(z0, z1)) 16.79/6.87
D(g(g(0, z0), z1), s(z2)) → c1(G(e(z0), d(g(g(0, z0), z1), z2)), D(g(g(0, z0), z1), z2), G(g(0, z0), z1), G(0, z0)) 16.79/6.87
D(g(z0, z1), z2) → c4(G(d(z0, z2), e(z1)), D(z0, z2)) 16.79/6.87
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:

c, c1, c4, c5

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

Removed 2 trailing tuple parts
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16.79/6.87

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(e(z0), z1) → h(d(z0, z1), s(z1)) 16.79/6.87
d(g(g(0, z0), z1), s(z2)) → g(e(z0), d(g(g(0, z0), z1), z2)) 16.79/6.87
d(g(g(0, z0), z1), 0) → e(z1) 16.79/6.87
d(g(0, z0), z1) → e(z0) 16.79/6.87
d(g(z0, z1), z2) → g(d(z0, z2), e(z1)) 16.79/6.87
g(e(z0), e(z1)) → e(g(z0, z1))
Tuples:

H(e(z0), z1) → c(H(d(z0, z1), s(z1)), D(z0, z1)) 16.79/6.87
D(g(z0, z1), z2) → c4(G(d(z0, z2), e(z1)), D(z0, z2)) 16.79/6.87
G(e(z0), e(z1)) → c5(G(z0, z1)) 16.79/6.87
D(g(g(0, z0), z1), s(z2)) → c1(G(e(z0), d(g(g(0, z0), z1), z2)), D(g(g(0, z0), z1), z2))
S tuples:

H(e(z0), z1) → c(H(d(z0, z1), s(z1)), D(z0, z1)) 16.79/6.87
D(g(z0, z1), z2) → c4(G(d(z0, z2), e(z1)), D(z0, z2)) 16.79/6.87
G(e(z0), e(z1)) → c5(G(z0, z1)) 16.79/6.87
D(g(g(0, z0), z1), s(z2)) → c1(G(e(z0), d(g(g(0, z0), z1), z2)), D(g(g(0, z0), z1), z2))
K tuples:none
Defined Rule Symbols:

h, d, g

Defined Pair Symbols:

H, D, G

Compound Symbols:

c, c4, c5, c1

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

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

H(e(g(g(0, z0), z1)), s(z2)) → c(H(g(e(z0), d(g(g(0, z0), z1), z2)), s(s(z2))), D(g(g(0, z0), z1), s(z2))) 16.79/6.87
H(e(g(g(0, z0), z1)), 0) → c(H(e(z1), s(0)), D(g(g(0, z0), z1), 0)) 16.79/6.87
H(e(g(0, z0)), z1) → c(H(e(z0), s(z1)), D(g(0, z0), z1)) 16.79/6.87
H(e(g(z0, z1)), z2) → c(H(g(d(z0, z2), e(z1)), s(z2)), D(g(z0, z1), z2))
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16.79/6.87

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(e(z0), z1) → h(d(z0, z1), s(z1)) 16.79/6.87
d(g(g(0, z0), z1), s(z2)) → g(e(z0), d(g(g(0, z0), z1), z2)) 16.79/6.87
d(g(g(0, z0), z1), 0) → e(z1) 16.79/6.87
d(g(0, z0), z1) → e(z0) 16.79/6.87
d(g(z0, z1), z2) → g(d(z0, z2), e(z1)) 16.79/6.87
g(e(z0), e(z1)) → e(g(z0, z1))
Tuples:

D(g(z0, z1), z2) → c4(G(d(z0, z2), e(z1)), D(z0, z2)) 16.79/6.87
G(e(z0), e(z1)) → c5(G(z0, z1)) 16.79/6.87
D(g(g(0, z0), z1), s(z2)) → c1(G(e(z0), d(g(g(0, z0), z1), z2)), D(g(g(0, z0), z1), z2)) 16.79/6.87
H(e(g(g(0, z0), z1)), s(z2)) → c(H(g(e(z0), d(g(g(0, z0), z1), z2)), s(s(z2))), D(g(g(0, z0), z1), s(z2))) 16.79/6.87
H(e(g(g(0, z0), z1)), 0) → c(H(e(z1), s(0)), D(g(g(0, z0), z1), 0)) 16.79/6.87
H(e(g(0, z0)), z1) → c(H(e(z0), s(z1)), D(g(0, z0), z1)) 16.79/6.87
H(e(g(z0, z1)), z2) → c(H(g(d(z0, z2), e(z1)), s(z2)), D(g(z0, z1), z2))
S tuples:

D(g(z0, z1), z2) → c4(G(d(z0, z2), e(z1)), D(z0, z2)) 16.79/6.87
G(e(z0), e(z1)) → c5(G(z0, z1)) 16.79/6.87
D(g(g(0, z0), z1), s(z2)) → c1(G(e(z0), d(g(g(0, z0), z1), z2)), D(g(g(0, z0), z1), z2)) 16.79/6.87
H(e(g(g(0, z0), z1)), s(z2)) → c(H(g(e(z0), d(g(g(0, z0), z1), z2)), s(s(z2))), D(g(g(0, z0), z1), s(z2))) 16.79/6.87
H(e(g(g(0, z0), z1)), 0) → c(H(e(z1), s(0)), D(g(g(0, z0), z1), 0)) 16.79/6.87
H(e(g(0, z0)), z1) → c(H(e(z0), s(z1)), D(g(0, z0), z1)) 16.79/6.87
H(e(g(z0, z1)), z2) → c(H(g(d(z0, z2), e(z1)), s(z2)), D(g(z0, z1), z2))
K tuples:none
Defined Rule Symbols:

h, d, g

Defined Pair Symbols:

D, G, H

Compound Symbols:

c4, c5, c1, c

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(7) CdtUnreachableProof (EQUIVALENT transformation)

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

D(g(z0, z1), z2) → c4(G(d(z0, z2), e(z1)), D(z0, z2)) 16.79/6.87
D(g(g(0, z0), z1), s(z2)) → c1(G(e(z0), d(g(g(0, z0), z1), z2)), D(g(g(0, z0), z1), z2)) 16.79/6.87
H(e(g(g(0, z0), z1)), s(z2)) → c(H(g(e(z0), d(g(g(0, z0), z1), z2)), s(s(z2))), D(g(g(0, z0), z1), s(z2))) 16.79/6.87
H(e(g(g(0, z0), z1)), 0) → c(H(e(z1), s(0)), D(g(g(0, z0), z1), 0)) 16.79/6.87
H(e(g(0, z0)), z1) → c(H(e(z0), s(z1)), D(g(0, z0), z1)) 16.79/6.87
H(e(g(z0, z1)), z2) → c(H(g(d(z0, z2), e(z1)), s(z2)), D(g(z0, z1), z2))
16.79/6.87
16.79/6.87

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(e(z0), z1) → h(d(z0, z1), s(z1)) 16.79/6.87
d(g(g(0, z0), z1), s(z2)) → g(e(z0), d(g(g(0, z0), z1), z2)) 16.79/6.87
d(g(g(0, z0), z1), 0) → e(z1) 16.79/6.87
d(g(0, z0), z1) → e(z0) 16.79/6.87
d(g(z0, z1), z2) → g(d(z0, z2), e(z1)) 16.79/6.87
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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(9) 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 : 16.79/6.87

POL(G(x1, x2)) = x1    16.79/6.87
POL(c5(x1)) = x1    16.79/6.87
POL(e(x1)) = [2] + x1   
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(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(e(z0), z1) → h(d(z0, z1), s(z1)) 16.79/6.87
d(g(g(0, z0), z1), s(z2)) → g(e(z0), d(g(g(0, z0), z1), z2)) 16.79/6.87
d(g(g(0, z0), z1), 0) → e(z1) 16.79/6.87
d(g(0, z0), z1) → e(z0) 16.79/6.87
d(g(z0, z1), z2) → g(d(z0, z2), e(z1)) 16.79/6.87
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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16.79/6.87

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

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

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17.13/6.95 EOF