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