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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(c(X, s(Y))) → f(c(s(X), Y))
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g(c(s(X), Y)) → f(c(X, s(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:
f(c(z0, s(z1))) → f(c(s(z0), z1))
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g(c(s(z0), z1)) → f(c(z0, s(z1)))
Tuples:
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
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G(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
S tuples:
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
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G(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F, G
Compound Symbols:
c1, c2
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(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
G(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
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(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(c(z0, s(z1))) → f(c(s(z0), z1))
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g(c(s(z0), z1)) → f(c(z0, s(z1)))
Tuples:
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
S tuples:
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F
Compound Symbols:
c1
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(5) 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(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
We considered the (Usable) Rules:none
And the Tuples:
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(F(x1)) = x1
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POL(c(x1, x2)) = x2
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POL(c1(x1)) = x1
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POL(s(x1)) = [2] + x1
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(c(z0, s(z1))) → f(c(s(z0), z1))
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g(c(s(z0), z1)) → f(c(z0, s(z1)))
Tuples:
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
S tuples:none
K tuples:
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F
Compound Symbols:
c1
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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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