YES(O(1), O(n^2)) 0.00/0.99 YES(O(1), O(n^2)) 0.00/1.00 0.00/1.00 0.00/1.00 0.00/1.00 0.00/1.00 0.00/1.00 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/1.00 0.00/1.00 0.00/1.00
0.00/1.00
0.00/1.00
0.00/1.00
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
0.00/1.00 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
0.00/1.00 
↳2 CdtProblem
0.00/1.00 
↳3 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/1.00 
↳4 CdtProblem
0.00/1.00 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
0.00/1.00 
↳6 CdtProblem
0.00/1.00 
↳7 SIsEmptyProof (BOTH BOUNDS(ID, ID))
0.00/1.00 
↳8 BOUNDS(O(1), O(1))
0.00/1.00
0.00/1.00 0.00/1.00
0.00/1.00
0.00/1.00

(0) Obligation:

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

sum(0) → 0 0.00/1.00
sum(s(x)) → +(sum(x), s(x)) 0.00/1.00
+(x, 0) → x 0.00/1.00
+(x, s(y)) → s(+(x, y))

Rewrite Strategy: INNERMOST
0.00/1.00
0.00/1.00

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
0.00/1.00
0.00/1.00

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

sum(0) → 0 0.00/1.00
sum(s(z0)) → +(sum(z0), s(z0)) 0.00/1.00
+(z0, 0) → z0 0.00/1.00
+(z0, s(z1)) → s(+(z0, z1))
Tuples:

SUM(s(z0)) → c1(+'(sum(z0), s(z0)), SUM(z0)) 0.00/1.00
+'(z0, s(z1)) → c3(+'(z0, z1))
S tuples:

SUM(s(z0)) → c1(+'(sum(z0), s(z0)), SUM(z0)) 0.00/1.00
+'(z0, s(z1)) → c3(+'(z0, z1))
K tuples:none
Defined Rule Symbols:

sum, +

Defined Pair Symbols:

SUM, +'

Compound Symbols:

c1, c3

0.00/1.00
0.00/1.00

(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.

SUM(s(z0)) → c1(+'(sum(z0), s(z0)), SUM(z0))
We considered the (Usable) Rules:

sum(0) → 0 0.00/1.00
sum(s(z0)) → +(sum(z0), s(z0)) 0.00/1.00
+(z0, s(z1)) → s(+(z0, z1)) 0.00/1.00
+(z0, 0) → z0
And the Tuples:

SUM(s(z0)) → c1(+'(sum(z0), s(z0)), SUM(z0)) 0.00/1.00
+'(z0, s(z1)) → c3(+'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/1.00

POL(+(x1, x2)) = [5]    0.00/1.00
POL(+'(x1, x2)) = [3]    0.00/1.00
POL(0) = [3]    0.00/1.00
POL(SUM(x1)) = x1    0.00/1.00
POL(c1(x1, x2)) = x1 + x2    0.00/1.00
POL(c3(x1)) = x1    0.00/1.00
POL(s(x1)) = [4] + x1    0.00/1.00
POL(sum(x1)) = 0   
0.00/1.00
0.00/1.00

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

sum(0) → 0 0.00/1.00
sum(s(z0)) → +(sum(z0), s(z0)) 0.00/1.00
+(z0, 0) → z0 0.00/1.00
+(z0, s(z1)) → s(+(z0, z1))
Tuples:

SUM(s(z0)) → c1(+'(sum(z0), s(z0)), SUM(z0)) 0.00/1.00
+'(z0, s(z1)) → c3(+'(z0, z1))
S tuples:

+'(z0, s(z1)) → c3(+'(z0, z1))
K tuples:

SUM(s(z0)) → c1(+'(sum(z0), s(z0)), SUM(z0))
Defined Rule Symbols:

sum, +

Defined Pair Symbols:

SUM, +'

Compound Symbols:

c1, c3

0.00/1.00
0.00/1.00

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

+'(z0, s(z1)) → c3(+'(z0, z1))
We considered the (Usable) Rules:

sum(0) → 0 0.00/1.00
sum(s(z0)) → +(sum(z0), s(z0)) 0.00/1.00
+(z0, s(z1)) → s(+(z0, z1)) 0.00/1.00
+(z0, 0) → z0
And the Tuples:

SUM(s(z0)) → c1(+'(sum(z0), s(z0)), SUM(z0)) 0.00/1.00
+'(z0, s(z1)) → c3(+'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/1.00

POL(+(x1, x2)) = [3] + [3]x2    0.00/1.00
POL(+'(x1, x2)) = x2    0.00/1.00
POL(0) = [1]    0.00/1.00
POL(SUM(x1)) = x12    0.00/1.00
POL(c1(x1, x2)) = x1 + x2    0.00/1.00
POL(c3(x1)) = x1    0.00/1.00
POL(s(x1)) = [1] + x1    0.00/1.00
POL(sum(x1)) = 0   
0.00/1.00
0.00/1.00

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

sum(0) → 0 0.00/1.00
sum(s(z0)) → +(sum(z0), s(z0)) 0.00/1.00
+(z0, 0) → z0 0.00/1.00
+(z0, s(z1)) → s(+(z0, z1))
Tuples:

SUM(s(z0)) → c1(+'(sum(z0), s(z0)), SUM(z0)) 0.00/1.00
+'(z0, s(z1)) → c3(+'(z0, z1))
S tuples:none
K tuples:

SUM(s(z0)) → c1(+'(sum(z0), s(z0)), SUM(z0)) 0.00/1.00
+'(z0, s(z1)) → c3(+'(z0, z1))
Defined Rule Symbols:

sum, +

Defined Pair Symbols:

SUM, +'

Compound Symbols:

c1, c3

0.00/1.00
0.00/1.00

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

The set S is empty
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0.00/1.00

(8) BOUNDS(O(1), O(1))

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0.00/1.00
2.39/1.07 EOF