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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
prime(0) → false
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prime(s(0)) → false
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prime(s(s(x))) → prime1(s(s(x)), s(x))
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prime1(x, 0) → false
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prime1(x, s(0)) → true
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prime1(x, s(s(y))) → and(not(divp(s(s(y)), x)), prime1(x, s(y)))
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divp(x, y) → =(rem(x, y), 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:
prime(0) → false
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prime(s(0)) → false
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prime(s(s(z0))) → prime1(s(s(z0)), s(z0))
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prime1(z0, 0) → false
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prime1(z0, s(0)) → true
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prime1(z0, s(s(z1))) → and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
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divp(z0, z1) → =(rem(z0, z1), 0)
Tuples:
PRIME(s(s(z0))) → c2(PRIME1(s(s(z0)), s(z0)))
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PRIME1(z0, s(s(z1))) → c5(DIVP(s(s(z1)), z0), PRIME1(z0, s(z1)))
S tuples:
PRIME(s(s(z0))) → c2(PRIME1(s(s(z0)), s(z0)))
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PRIME1(z0, s(s(z1))) → c5(DIVP(s(s(z1)), z0), PRIME1(z0, s(z1)))
K tuples:none
Defined Rule Symbols:
prime, prime1, divp
Defined Pair Symbols:
PRIME, PRIME1
Compound Symbols:
c2, c5
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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:
prime(0) → false
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prime(s(0)) → false
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prime(s(s(z0))) → prime1(s(s(z0)), s(z0))
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prime1(z0, 0) → false
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prime1(z0, s(0)) → true
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prime1(z0, s(s(z1))) → and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
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divp(z0, z1) → =(rem(z0, z1), 0)
Tuples:
PRIME(s(s(z0))) → c2(PRIME1(s(s(z0)), s(z0)))
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PRIME1(z0, s(s(z1))) → c5(PRIME1(z0, s(z1)))
S tuples:
PRIME(s(s(z0))) → c2(PRIME1(s(s(z0)), s(z0)))
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PRIME1(z0, s(s(z1))) → c5(PRIME1(z0, s(z1)))
K tuples:none
Defined Rule Symbols:
prime, prime1, divp
Defined Pair Symbols:
PRIME, PRIME1
Compound Symbols:
c2, c5
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(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
PRIME(s(s(z0))) → c2(PRIME1(s(s(z0)), s(z0)))
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
prime(0) → false
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prime(s(0)) → false
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prime(s(s(z0))) → prime1(s(s(z0)), s(z0))
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prime1(z0, 0) → false
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prime1(z0, s(0)) → true
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prime1(z0, s(s(z1))) → and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
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divp(z0, z1) → =(rem(z0, z1), 0)
Tuples:
PRIME1(z0, s(s(z1))) → c5(PRIME1(z0, s(z1)))
S tuples:
PRIME1(z0, s(s(z1))) → c5(PRIME1(z0, s(z1)))
K tuples:none
Defined Rule Symbols:
prime, prime1, divp
Defined Pair Symbols:
PRIME1
Compound Symbols:
c5
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(7) 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.
PRIME1(z0, s(s(z1))) → c5(PRIME1(z0, s(z1)))
We considered the (Usable) Rules:none
And the Tuples:
PRIME1(z0, s(s(z1))) → c5(PRIME1(z0, s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(PRIME1(x1, x2)) = x2
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POL(c5(x1)) = x1
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POL(s(x1)) = [1] + x1
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(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
prime(0) → false
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prime(s(0)) → false
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prime(s(s(z0))) → prime1(s(s(z0)), s(z0))
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prime1(z0, 0) → false
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prime1(z0, s(0)) → true
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prime1(z0, s(s(z1))) → and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
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divp(z0, z1) → =(rem(z0, z1), 0)
Tuples:
PRIME1(z0, s(s(z1))) → c5(PRIME1(z0, s(z1)))
S tuples:none
K tuples:
PRIME1(z0, s(s(z1))) → c5(PRIME1(z0, s(z1)))
Defined Rule Symbols:
prime, prime1, divp
Defined Pair Symbols:
PRIME1
Compound Symbols:
c5
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(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
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(10) BOUNDS(O(1), O(1))
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