YES(O(1), O(n^1)) 0.00/0.80 YES(O(1), O(n^1)) 0.00/0.81 0.00/0.81 0.00/0.81 0.00/0.81 0.00/0.81 0.00/0.81 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.81 0.00/0.81 0.00/0.81
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
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↳2 CdtProblem
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↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳4 CdtProblem
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↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
0.00/0.81 
↳6 CdtProblem
0.00/0.81 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.81 
↳8 CdtProblem
0.00/0.81 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.81 
↳10 CdtProblem
0.00/0.81 
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳12 BOUNDS(O(1), O(1))
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0.00/0.81 0.00/0.81
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0.00/0.81

(0) Obligation:

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

merge(x, nil) → x 0.00/0.81
merge(nil, y) → y 0.00/0.81
merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v))) 0.00/0.81
merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v))

Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
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0.00/0.81

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

merge(z0, nil) → z0 0.00/0.81
merge(nil, z0) → z0 0.00/0.81
merge(++(z0, z1), ++(u, v)) → ++(z0, merge(z1, ++(u, v))) 0.00/0.81
merge(++(z0, z1), ++(u, v)) → ++(u, merge(++(z0, z1), v))
Tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v))) 0.00/0.81
MERGE(++(z0, z1), ++(u, v)) → c3(MERGE(++(z0, z1), v))
S tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v))) 0.00/0.81
MERGE(++(z0, z1), ++(u, v)) → c3(MERGE(++(z0, z1), v))
K tuples:none
Defined Rule Symbols:

merge

Defined Pair Symbols:

MERGE

Compound Symbols:

c2, c3

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(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
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0.00/0.81

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

merge(z0, nil) → z0 0.00/0.81
merge(nil, z0) → z0 0.00/0.81
merge(++(z0, z1), ++(u, v)) → ++(z0, merge(z1, ++(u, v))) 0.00/0.81
merge(++(z0, z1), ++(u, v)) → ++(u, merge(++(z0, z1), v))
Tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v))) 0.00/0.81
MERGE(++(z0, z1), ++(u, v)) → c3
S tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v))) 0.00/0.81
MERGE(++(z0, z1), ++(u, v)) → c3
K tuples:none
Defined Rule Symbols:

merge

Defined Pair Symbols:

MERGE

Compound Symbols:

c2, c3

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(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

MERGE(++(z0, z1), ++(u, v)) → c3
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0.00/0.81

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

merge(z0, nil) → z0 0.00/0.81
merge(nil, z0) → z0 0.00/0.81
merge(++(z0, z1), ++(u, v)) → ++(z0, merge(z1, ++(u, v))) 0.00/0.81
merge(++(z0, z1), ++(u, v)) → ++(u, merge(++(z0, z1), v))
Tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v))) 0.00/0.81
MERGE(++(z0, z1), ++(u, v)) → c3
S tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v))) 0.00/0.81
MERGE(++(z0, z1), ++(u, v)) → c3
K tuples:none
Defined Rule Symbols:

merge

Defined Pair Symbols:

MERGE

Compound Symbols:

c2, c3

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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.

MERGE(++(z0, z1), ++(u, v)) → c3
We considered the (Usable) Rules:none
And the Tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v))) 0.00/0.81
MERGE(++(z0, z1), ++(u, v)) → c3
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.81

POL(++(x1, x2)) = [1]    0.00/0.81
POL(MERGE(x1, x2)) = [2]x2    0.00/0.81
POL(c2(x1)) = x1    0.00/0.81
POL(c3) = 0    0.00/0.81
POL(u) = [1]    0.00/0.81
POL(v) = [1]   
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

merge(z0, nil) → z0 0.00/0.81
merge(nil, z0) → z0 0.00/0.81
merge(++(z0, z1), ++(u, v)) → ++(z0, merge(z1, ++(u, v))) 0.00/0.81
merge(++(z0, z1), ++(u, v)) → ++(u, merge(++(z0, z1), v))
Tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v))) 0.00/0.81
MERGE(++(z0, z1), ++(u, v)) → c3
S tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v)))
K tuples:

MERGE(++(z0, z1), ++(u, v)) → c3
Defined Rule Symbols:

merge

Defined Pair Symbols:

MERGE

Compound Symbols:

c2, c3

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0.00/0.81

(9) 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.

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v)))
We considered the (Usable) Rules:none
And the Tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v))) 0.00/0.81
MERGE(++(z0, z1), ++(u, v)) → c3
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.81

POL(++(x1, x2)) = [1] + x2    0.00/0.81
POL(MERGE(x1, x2)) = [3]x1    0.00/0.81
POL(c2(x1)) = x1    0.00/0.81
POL(c3) = 0    0.00/0.81
POL(u) = 0    0.00/0.81
POL(v) = 0   
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0.00/0.81

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

merge(z0, nil) → z0 0.00/0.81
merge(nil, z0) → z0 0.00/0.81
merge(++(z0, z1), ++(u, v)) → ++(z0, merge(z1, ++(u, v))) 0.00/0.81
merge(++(z0, z1), ++(u, v)) → ++(u, merge(++(z0, z1), v))
Tuples:

MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v))) 0.00/0.81
MERGE(++(z0, z1), ++(u, v)) → c3
S tuples:none
K tuples:

MERGE(++(z0, z1), ++(u, v)) → c3 0.00/0.81
MERGE(++(z0, z1), ++(u, v)) → c2(MERGE(z1, ++(u, v)))
Defined Rule Symbols:

merge

Defined Pair Symbols:

MERGE

Compound Symbols:

c2, c3

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(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty
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(12) BOUNDS(O(1), O(1))

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0.00/0.84 EOF