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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
odd(S(x)) → even(x)
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even(S(x)) → odd(x)
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odd(0) → 0
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even(0) → S(0)
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:
odd(S(z0)) → even(z0)
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odd(0) → 0
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even(S(z0)) → odd(z0)
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even(0) → S(0)
Tuples:
ODD(S(z0)) → c(EVEN(z0))
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EVEN(S(z0)) → c2(ODD(z0))
S tuples:
ODD(S(z0)) → c(EVEN(z0))
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EVEN(S(z0)) → c2(ODD(z0))
K tuples:none
Defined Rule Symbols:
odd, even
Defined Pair Symbols:
ODD, EVEN
Compound Symbols:
c, c2
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(3) 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.
ODD(S(z0)) → c(EVEN(z0))
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EVEN(S(z0)) → c2(ODD(z0))
We considered the (Usable) Rules:none
And the Tuples:
ODD(S(z0)) → c(EVEN(z0))
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EVEN(S(z0)) → c2(ODD(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(EVEN(x1)) = [2] + [2]x1
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POL(ODD(x1)) = [4] + [2]x1
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POL(S(x1)) = [4] + x1
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POL(c(x1)) = x1
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POL(c2(x1)) = x1
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(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
odd(S(z0)) → even(z0)
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odd(0) → 0
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even(S(z0)) → odd(z0)
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even(0) → S(0)
Tuples:
ODD(S(z0)) → c(EVEN(z0))
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EVEN(S(z0)) → c2(ODD(z0))
S tuples:none
K tuples:
ODD(S(z0)) → c(EVEN(z0))
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EVEN(S(z0)) → c2(ODD(z0))
Defined Rule Symbols:
odd, even
Defined Pair Symbols:
ODD, EVEN
Compound Symbols:
c, c2
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(5) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
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(6) BOUNDS(O(1), O(1))
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