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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
p(s(x)) → x
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fac(0) → s(0)
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fac(s(x)) → times(s(x), fac(p(s(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:
p(s(z0)) → z0
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fac(0) → s(0)
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fac(s(z0)) → times(s(z0), fac(p(s(z0))))
Tuples:
FAC(s(z0)) → c2(FAC(p(s(z0))), P(s(z0)))
S tuples:
FAC(s(z0)) → c2(FAC(p(s(z0))), P(s(z0)))
K tuples:none
Defined Rule Symbols:
p, fac
Defined Pair Symbols:
FAC
Compound Symbols:
c2
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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:
p(s(z0)) → z0
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fac(0) → s(0)
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fac(s(z0)) → times(s(z0), fac(p(s(z0))))
Tuples:
FAC(s(z0)) → c2(FAC(p(s(z0))))
S tuples:
FAC(s(z0)) → c2(FAC(p(s(z0))))
K tuples:none
Defined Rule Symbols:
p, fac
Defined Pair Symbols:
FAC
Compound Symbols:
c2
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(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
FAC(
s(
z0)) →
c2(
FAC(
p(
s(
z0)))) by
FAC(s(z0)) → c2(FAC(z0))
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
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fac(0) → s(0)
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fac(s(z0)) → times(s(z0), fac(p(s(z0))))
Tuples:
FAC(s(z0)) → c2(FAC(z0))
S tuples:
FAC(s(z0)) → c2(FAC(z0))
K tuples:none
Defined Rule Symbols:
p, fac
Defined Pair Symbols:
FAC
Compound Symbols:
c2
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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.
FAC(s(z0)) → c2(FAC(z0))
We considered the (Usable) Rules:none
And the Tuples:
FAC(s(z0)) → c2(FAC(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(FAC(x1)) = [3]x1
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POL(c2(x1)) = x1
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POL(s(x1)) = [1] + x1
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(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
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fac(0) → s(0)
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fac(s(z0)) → times(s(z0), fac(p(s(z0))))
Tuples:
FAC(s(z0)) → c2(FAC(z0))
S tuples:none
K tuples:
FAC(s(z0)) → c2(FAC(z0))
Defined Rule Symbols:
p, fac
Defined Pair Symbols:
FAC
Compound Symbols:
c2
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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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