YES(O(1), O(1)) 0.00/0.54 YES(O(1), O(1)) 0.00/0.55 0.00/0.55 0.00/0.55 0.00/0.55 0.00/0.55 0.00/0.55 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.55 0.00/0.55 0.00/0.55
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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 CdtUnreachableProof (⇔)
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↳4 CdtProblem
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↳5 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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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:

+(x, 0) → x 0.00/0.55
+(x, i(x)) → 0 0.00/0.55
+(+(x, y), z) → +(x, +(y, z)) 0.00/0.55
*(x, +(y, z)) → +(*(x, y), *(x, z)) 0.00/0.55
*(+(x, y), z) → +(*(x, z), *(y, z))

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:

+(z0, 0) → z0 0.00/0.55
+(z0, i(z0)) → 0 0.00/0.55
+(+(z0, z1), z2) → +(z0, +(z1, z2)) 0.00/0.55
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2)) 0.00/0.55
*(+(z0, z1), z2) → +(*(z0, z2), *(z1, z2))
Tuples:

+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2)) 0.00/0.55
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2)) 0.00/0.55
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))
S tuples:

+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2)) 0.00/0.55
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2)) 0.00/0.55
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))
K tuples:none
Defined Rule Symbols:

+, *

Defined Pair Symbols:

+', *'

Compound Symbols:

c2, c3, c4

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(3) CdtUnreachableProof (EQUIVALENT transformation)

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

+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2)) 0.00/0.55
*'(z0, +(z1, z2)) → c3(+'(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2)) 0.00/0.55
*'(+(z0, z1), z2) → c4(+'(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(z0, 0) → z0 0.00/0.55
+(z0, i(z0)) → 0 0.00/0.55
+(+(z0, z1), z2) → +(z0, +(z1, z2)) 0.00/0.55
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2)) 0.00/0.55
*(+(z0, z1), z2) → +(*(z0, z2), *(z1, z2))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

+, *

Defined Pair Symbols:none

Compound Symbols:none

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

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

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0.00/0.58 EOF