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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(s(X), X) → f(X, a(X))
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f(X, c(X)) → f(s(X), X)
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f(X, X) → c(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), z0) → f(z0, a(z0))
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f(z0, c(z0)) → f(s(z0), z0)
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f(z0, z0) → c(z0)
Tuples:
F(s(z0), z0) → c1(F(z0, a(z0)))
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F(z0, c(z0)) → c2(F(s(z0), z0))
S tuples:
F(s(z0), z0) → c1(F(z0, a(z0)))
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F(z0, c(z0)) → c2(F(s(z0), z0))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c1, 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:
f(s(z0), z0) → f(z0, a(z0))
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f(z0, c(z0)) → f(s(z0), z0)
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f(z0, z0) → c(z0)
Tuples:
F(z0, c(z0)) → c2(F(s(z0), z0))
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F(s(z0), z0) → c1
S tuples:
F(z0, c(z0)) → c2(F(s(z0), z0))
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F(s(z0), z0) → c1
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c2, c1
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(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
F(s(z0), z0) → c1
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F(z0, c(z0)) → c2(F(s(z0), z0))
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(s(z0), z0) → f(z0, a(z0))
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f(z0, c(z0)) → f(s(z0), z0)
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f(z0, z0) → c(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f
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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