YES(O(1), O(n^1)) 0.00/0.75 YES(O(1), O(n^1)) 0.00/0.76 0.00/0.76 0.00/0.76 0.00/0.76 0.00/0.76 0.00/0.76 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.76 0.00/0.76 0.00/0.76
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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 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳4 CdtProblem
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↳5 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳6 BOUNDS(O(1), O(1))
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(0) Obligation:

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

average(s(x), y) → average(x, s(y)) 0.00/0.76
average(x, s(s(s(y)))) → s(average(s(x), y)) 0.00/0.76
average(0, 0) → 0 0.00/0.76
average(0, s(0)) → 0 0.00/0.76
average(0, s(s(0))) → s(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:

average(s(z0), z1) → average(z0, s(z1)) 0.00/0.76
average(z0, s(s(s(z1)))) → s(average(s(z0), z1)) 0.00/0.76
average(0, 0) → 0 0.00/0.76
average(0, s(0)) → 0 0.00/0.76
average(0, s(s(0))) → s(0)
Tuples:

AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1))) 0.00/0.76
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
S tuples:

AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1))) 0.00/0.76
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
K tuples:none
Defined Rule Symbols:

average

Defined Pair Symbols:

AVERAGE

Compound Symbols:

c, c1

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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.

AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1))) 0.00/0.76
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
We considered the (Usable) Rules:none
And the Tuples:

AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1))) 0.00/0.76
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.76

POL(AVERAGE(x1, x2)) = [4]x1 + [2]x2    0.00/0.76
POL(c(x1)) = x1    0.00/0.76
POL(c1(x1)) = x1    0.00/0.76
POL(s(x1)) = [1] + x1   
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

average(s(z0), z1) → average(z0, s(z1)) 0.00/0.76
average(z0, s(s(s(z1)))) → s(average(s(z0), z1)) 0.00/0.76
average(0, 0) → 0 0.00/0.76
average(0, s(0)) → 0 0.00/0.76
average(0, s(s(0))) → s(0)
Tuples:

AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1))) 0.00/0.76
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
S tuples:none
K tuples:

AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1))) 0.00/0.76
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
Defined Rule Symbols:

average

Defined Pair Symbols:

AVERAGE

Compound Symbols:

c, c1

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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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0.00/0.79 EOF