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))
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↳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:

terms(N) → cons(recip(sqr(N))) 0.00/0.55
sqr(0) → 0 0.00/0.55
sqr(s) → s 0.00/0.55
dbl(0) → 0 0.00/0.55
dbl(s) → s 0.00/0.55
add(0, X) → X 0.00/0.55
add(s, Y) → s 0.00/0.55
first(0, X) → nil 0.00/0.55
first(s, cons(Y)) → cons(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:

terms(z0) → cons(recip(sqr(z0))) 0.00/0.55
sqr(0) → 0 0.00/0.55
sqr(s) → s 0.00/0.55
dbl(0) → 0 0.00/0.55
dbl(s) → s 0.00/0.55
add(0, z0) → z0 0.00/0.55
add(s, z0) → s 0.00/0.55
first(0, z0) → nil 0.00/0.55
first(s, cons(z0)) → cons(z0)
Tuples:

TERMS(z0) → c(SQR(z0))
S tuples:

TERMS(z0) → c(SQR(z0))
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first

Defined Pair Symbols:

TERMS

Compound Symbols:

c

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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:

terms(z0) → cons(recip(sqr(z0))) 0.00/0.55
sqr(0) → 0 0.00/0.55
sqr(s) → s 0.00/0.55
dbl(0) → 0 0.00/0.55
dbl(s) → s 0.00/0.55
add(0, z0) → z0 0.00/0.55
add(s, z0) → s 0.00/0.55
first(0, z0) → nil 0.00/0.55
first(s, cons(z0)) → cons(z0)
Tuples:

TERMS(z0) → c
S tuples:

TERMS(z0) → c
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first

Defined Pair Symbols:

TERMS

Compound Symbols:

c

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

Removed 1 trailing nodes:

TERMS(z0) → c
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0))) 0.00/0.55
sqr(0) → 0 0.00/0.55
sqr(s) → s 0.00/0.55
dbl(0) → 0 0.00/0.55
dbl(s) → s 0.00/0.55
add(0, z0) → z0 0.00/0.55
add(s, z0) → s 0.00/0.55
first(0, z0) → nil 0.00/0.55
first(s, cons(z0)) → cons(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first

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