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(0) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
add0(S(x), x2) → +(S(0), add0(x2, x))
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add0(0, x2) → x2
The (relative) TRS S consists of the following rules:
+(x, S(0)) → S(x)
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+(S(0), y) → S(y)
Rewrite Strategy: INNERMOST
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(1) CpxRelTrsToCDT (UPPER BOUND (ID) transformation)
Relative innermost TRS to CDT Problem.
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(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(z0, S(0)) → S(z0)
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+(S(0), z0) → S(z0)
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add0(S(z0), z1) → +(S(0), add0(z1, z0))
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add0(0, z0) → z0
Tuples:
ADD0(S(z0), z1) → c2(+'(S(0), add0(z1, z0)), ADD0(z1, z0))
S tuples:
ADD0(S(z0), z1) → c2(+'(S(0), add0(z1, z0)), ADD0(z1, z0))
K tuples:none
Defined Rule Symbols:
add0, +
Defined Pair Symbols:
ADD0
Compound Symbols:
c2
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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:
+(z0, S(0)) → S(z0)
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+(S(0), z0) → S(z0)
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add0(S(z0), z1) → +(S(0), add0(z1, z0))
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add0(0, z0) → z0
Tuples:
ADD0(S(z0), z1) → c2(ADD0(z1, z0))
S tuples:
ADD0(S(z0), z1) → c2(ADD0(z1, z0))
K tuples:none
Defined Rule Symbols:
add0, +
Defined Pair Symbols:
ADD0
Compound Symbols:
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.
ADD0(S(z0), z1) → c2(ADD0(z1, z0))
We considered the (Usable) Rules:none
And the Tuples:
ADD0(S(z0), z1) → c2(ADD0(z1, z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(ADD0(x1, x2)) = [5]x1 + [5]x2
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POL(S(x1)) = [1] + x1
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POL(c2(x1)) = x1
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(z0, S(0)) → S(z0)
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+(S(0), z0) → S(z0)
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add0(S(z0), z1) → +(S(0), add0(z1, z0))
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add0(0, z0) → z0
Tuples:
ADD0(S(z0), z1) → c2(ADD0(z1, z0))
S tuples:none
K tuples:
ADD0(S(z0), z1) → c2(ADD0(z1, z0))
Defined Rule Symbols:
add0, +
Defined Pair Symbols:
ADD0
Compound Symbols:
c2
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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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