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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(0) → cons(0)
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f(s(0)) → f(p(s(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(0) → cons(0)
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f(s(0)) → f(p(s(0)))
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p(s(z0)) → z0
Tuples:
F(s(0)) → c1(F(p(s(0))), P(s(0)))
S tuples:
F(s(0)) → c1(F(p(s(0))), P(s(0)))
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F
Compound Symbols:
c1
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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(0) → cons(0)
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f(s(0)) → f(p(s(0)))
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p(s(z0)) → z0
Tuples:
F(s(0)) → c1(F(p(s(0))))
S tuples:
F(s(0)) → c1(F(p(s(0))))
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F
Compound Symbols:
c1
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(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
s(
0)) →
c1(
F(
p(
s(
0)))) by
F(s(0)) → c1(F(0))
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → cons(0)
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f(s(0)) → f(p(s(0)))
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p(s(z0)) → z0
Tuples:
F(s(0)) → c1(F(0))
S tuples:
F(s(0)) → c1(F(0))
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F
Compound Symbols:
c1
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(7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
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(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → cons(0)
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f(s(0)) → f(p(s(0)))
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p(s(z0)) → z0
Tuples:
F(s(0)) → c1
S tuples:
F(s(0)) → c1
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F
Compound Symbols:
c1
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(9) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(s(0)) → c1
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(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → cons(0)
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f(s(0)) → f(p(s(0)))
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p(s(z0)) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:none
Compound Symbols:none
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(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
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(12) BOUNDS(O(1), O(1))
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