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 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳4 CdtProblem
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↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
0.00/0.55 
↳6 CdtProblem
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↳7 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳8 BOUNDS(O(1), O(1))
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(0) Obligation:

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

zeros → cons(0, n__zeros) 0.00/0.55
tail(cons(X, XS)) → activate(XS) 0.00/0.55
zeros → n__zeros 0.00/0.55
activate(n__zeros) → zeros 0.00/0.55
activate(X) → X

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:

zeros → cons(0, n__zeros) 0.00/0.55
zeros → n__zeros 0.00/0.55
tail(cons(z0, z1)) → activate(z1) 0.00/0.55
activate(n__zeros) → zeros 0.00/0.55
activate(z0) → z0
Tuples:

TAIL(cons(z0, z1)) → c2(ACTIVATE(z1)) 0.00/0.55
ACTIVATE(n__zeros) → c3(ZEROS)
S tuples:

TAIL(cons(z0, z1)) → c2(ACTIVATE(z1)) 0.00/0.55
ACTIVATE(n__zeros) → c3(ZEROS)
K tuples:none
Defined Rule Symbols:

zeros, tail, activate

Defined Pair Symbols:

TAIL, ACTIVATE

Compound Symbols:

c2, c3

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

Removed 1 trailing tuple parts
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

zeros → cons(0, n__zeros) 0.00/0.55
zeros → n__zeros 0.00/0.55
tail(cons(z0, z1)) → activate(z1) 0.00/0.55
activate(n__zeros) → zeros 0.00/0.55
activate(z0) → z0
Tuples:

TAIL(cons(z0, z1)) → c2(ACTIVATE(z1)) 0.00/0.55
ACTIVATE(n__zeros) → c3
S tuples:

TAIL(cons(z0, z1)) → c2(ACTIVATE(z1)) 0.00/0.55
ACTIVATE(n__zeros) → c3
K tuples:none
Defined Rule Symbols:

zeros, tail, activate

Defined Pair Symbols:

TAIL, ACTIVATE

Compound Symbols:

c2, c3

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

Removed 2 trailing nodes:

ACTIVATE(n__zeros) → c3 0.00/0.55
TAIL(cons(z0, z1)) → c2(ACTIVATE(z1))
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

zeros → cons(0, n__zeros) 0.00/0.55
zeros → n__zeros 0.00/0.55
tail(cons(z0, z1)) → activate(z1) 0.00/0.55
activate(n__zeros) → zeros 0.00/0.55
activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

zeros, tail, activate

Defined Pair Symbols:none

Compound Symbols:none

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

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

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0.00/0.58 EOF