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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
*(x, +(y, z)) → +(*(x, y), *(x, z))
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:
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
Tuples:
*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
S tuples:
*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
K tuples:none
Defined Rule Symbols:
*
Defined Pair Symbols:
*'
Compound Symbols:
c
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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.
*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:
*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(*'(x1, x2)) = x2
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POL(+(x1, x2)) = [1] + x1 + x2
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POL(c(x1, x2)) = x1 + x2
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(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
*(z0, +(z1, z2)) → +(*(z0, z1), *(z0, z2))
Tuples:
*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
S tuples:none
K tuples:
*'(z0, +(z1, z2)) → c(*'(z0, z1), *'(z0, z2))
Defined Rule Symbols:
*
Defined Pair Symbols:
*'
Compound Symbols:
c
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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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