YES(O(1), O(n^1)) 0.00/0.71 YES(O(1), O(n^1)) 0.00/0.73 0.00/0.73 0.00/0.73 0.00/0.73 0.00/0.73 0.00/0.73 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.73 0.00/0.73 0.00/0.73
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(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

or(true, y) → true 0.00/0.73
or(x, true) → true 0.00/0.73
or(false, false) → false 0.00/0.73
mem(x, nil) → false 0.00/0.73
mem(x, set(y)) → =(x, y) 0.00/0.73
mem(x, union(y, z)) → or(mem(x, y), mem(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:

or(true, z0) → true 0.00/0.73
or(z0, true) → true 0.00/0.73
or(false, false) → false 0.00/0.73
mem(z0, nil) → false 0.00/0.73
mem(z0, set(z1)) → =(z0, z1) 0.00/0.73
mem(z0, union(z1, z2)) → or(mem(z0, z1), mem(z0, z2))
Tuples:

MEM(z0, union(z1, z2)) → c5(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z1), MEM(z0, z2))
S tuples:

MEM(z0, union(z1, z2)) → c5(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z1), MEM(z0, z2))
K tuples:none
Defined Rule Symbols:

or, mem

Defined Pair Symbols:

MEM

Compound Symbols:

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:

or(true, z0) → true 0.00/0.73
or(z0, true) → true 0.00/0.73
or(false, false) → false 0.00/0.73
mem(z0, nil) → false 0.00/0.73
mem(z0, set(z1)) → =(z0, z1) 0.00/0.73
mem(z0, union(z1, z2)) → or(mem(z0, z1), mem(z0, z2))
Tuples:

MEM(z0, union(z1, z2)) → c5(MEM(z0, z1), MEM(z0, z2))
S tuples:

MEM(z0, union(z1, z2)) → c5(MEM(z0, z1), MEM(z0, z2))
K tuples:none
Defined Rule Symbols:

or, mem

Defined Pair Symbols:

MEM

Compound Symbols:

c5

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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.

MEM(z0, union(z1, z2)) → c5(MEM(z0, z1), MEM(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:

MEM(z0, union(z1, z2)) → c5(MEM(z0, z1), MEM(z0, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.73

POL(MEM(x1, x2)) = x2    0.00/0.73
POL(c5(x1, x2)) = x1 + x2    0.00/0.73
POL(union(x1, x2)) = [1] + x1 + x2   
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

or(true, z0) → true 0.00/0.73
or(z0, true) → true 0.00/0.73
or(false, false) → false 0.00/0.73
mem(z0, nil) → false 0.00/0.73
mem(z0, set(z1)) → =(z0, z1) 0.00/0.73
mem(z0, union(z1, z2)) → or(mem(z0, z1), mem(z0, z2))
Tuples:

MEM(z0, union(z1, z2)) → c5(MEM(z0, z1), MEM(z0, z2))
S tuples:none
K tuples:

MEM(z0, union(z1, z2)) → c5(MEM(z0, z1), MEM(z0, z2))
Defined Rule Symbols:

or, mem

Defined Pair Symbols:

MEM

Compound Symbols:

c5

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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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0.00/0.78 EOF