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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
not(true) → false
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not(false) → true
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evenodd(x, 0) → not(evenodd(x, s(0)))
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evenodd(0, s(0)) → false
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evenodd(s(x), s(0)) → evenodd(x, 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:
not(true) → false
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not(false) → true
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evenodd(z0, 0) → not(evenodd(z0, s(0)))
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evenodd(0, s(0)) → false
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evenodd(s(z0), s(0)) → evenodd(z0, 0)
Tuples:
EVENODD(z0, 0) → c2(NOT(evenodd(z0, s(0))), EVENODD(z0, s(0)))
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EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
S tuples:
EVENODD(z0, 0) → c2(NOT(evenodd(z0, s(0))), EVENODD(z0, s(0)))
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EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
K tuples:none
Defined Rule Symbols:
not, evenodd
Defined Pair Symbols:
EVENODD
Compound Symbols:
c2, c4
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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:
not(true) → false
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not(false) → true
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evenodd(z0, 0) → not(evenodd(z0, s(0)))
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evenodd(0, s(0)) → false
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evenodd(s(z0), s(0)) → evenodd(z0, 0)
Tuples:
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
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EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
S tuples:
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
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EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
K tuples:none
Defined Rule Symbols:
not, evenodd
Defined Pair Symbols:
EVENODD
Compound Symbols:
c4, c2
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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.
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
We considered the (Usable) Rules:none
And the Tuples:
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
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EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(0) = 0
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POL(EVENODD(x1, x2)) = [2]x1
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POL(c2(x1)) = x1
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POL(c4(x1)) = x1
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POL(s(x1)) = [1] + x1
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
not(true) → false
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not(false) → true
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evenodd(z0, 0) → not(evenodd(z0, s(0)))
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evenodd(0, s(0)) → false
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evenodd(s(z0), s(0)) → evenodd(z0, 0)
Tuples:
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
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EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
S tuples:
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
K tuples:
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
Defined Rule Symbols:
not, evenodd
Defined Pair Symbols:
EVENODD
Compound Symbols:
c4, c2
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(7) CdtKnowledgeProof (EQUIVALENT transformation)
The following tuples could be moved from S to K by knowledge propagation:
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
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EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
Now S is empty
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(8) BOUNDS(O(1), O(1))
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