YES(O(1), O(n^1)) 2.51/1.08 YES(O(1), O(n^1)) 2.87/1.14 2.87/1.14 2.87/1.14 2.87/1.14 2.87/1.15 2.87/1.15 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 2.87/1.15 2.87/1.15 2.87/1.15
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2.87/1.15
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
2.87/1.15 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
2.87/1.15 
↳2 CdtProblem
2.87/1.15 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
2.87/1.15 
↳4 CdtProblem
2.87/1.15 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.87/1.15 
↳6 CdtProblem
2.87/1.15 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.87/1.15 
↳8 CdtProblem
2.87/1.15 
↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID))
2.87/1.15 
↳10 BOUNDS(O(1), O(1))
2.87/1.15
2.87/1.15 2.87/1.15
2.87/1.15
2.87/1.15

(0) Obligation:

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

+(*(x, y), *(x, z)) → *(x, +(y, z)) 2.87/1.15
+(+(x, y), z) → +(x, +(y, z)) 2.87/1.15
+(*(x, y), +(*(x, z), u)) → +(*(x, +(y, z)), u)

Rewrite Strategy: INNERMOST
2.87/1.15
2.87/1.15

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

Converted CpxTRS to CDT
2.87/1.15
2.87/1.15

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2)) 2.87/1.15
+(+(z0, z1), z2) → +(z0, +(z1, z2)) 2.87/1.15
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2)) 2.87/1.15
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2)) 2.87/1.15
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(*(z0, +(z1, z2)), u), +'(z1, z2))
S tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2)) 2.87/1.15
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2)) 2.87/1.15
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(*(z0, +(z1, z2)), u), +'(z1, z2))
K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2

2.87/1.15
2.87/1.15

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

Removed 1 trailing tuple parts
2.87/1.15
2.87/1.15

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2)) 2.87/1.15
+(+(z0, z1), z2) → +(z0, +(z1, z2)) 2.87/1.15
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2)) 2.87/1.15
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2)) 2.87/1.15
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(z1, z2))
S tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2)) 2.87/1.15
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2)) 2.87/1.15
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(z1, z2))
K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2

2.87/1.15
2.87/1.15

(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.

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2)) 2.87/1.15
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(z1, z2))
We considered the (Usable) Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2)) 2.87/1.15
+(+(z0, z1), z2) → +(z0, +(z1, z2)) 2.87/1.15
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
And the Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2)) 2.87/1.15
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2)) 2.87/1.15
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.87/1.15

POL(*(x1, x2)) = [1] + x2    2.87/1.15
POL(+(x1, x2)) = x1 + [2]x2    2.87/1.15
POL(+'(x1, x2)) = [2]x1    2.87/1.15
POL(c(x1)) = x1    2.87/1.15
POL(c1(x1, x2)) = x1 + x2    2.87/1.15
POL(c2(x1)) = x1    2.87/1.15
POL(u) = [3]   
2.87/1.15
2.87/1.15

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2)) 2.87/1.15
+(+(z0, z1), z2) → +(z0, +(z1, z2)) 2.87/1.15
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2)) 2.87/1.15
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2)) 2.87/1.15
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(z1, z2))
S tuples:

+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
K tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2)) 2.87/1.15
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(z1, z2))
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2

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2.87/1.15

(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.

+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
We considered the (Usable) Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2)) 2.87/1.15
+(+(z0, z1), z2) → +(z0, +(z1, z2)) 2.87/1.15
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
And the Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2)) 2.87/1.15
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2)) 2.87/1.15
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.87/1.15

POL(*(x1, x2)) = x1 + x2    2.87/1.15
POL(+(x1, x2)) = [4] + x1 + x2    2.87/1.15
POL(+'(x1, x2)) = [1] + [4]x1    2.87/1.15
POL(c(x1)) = x1    2.87/1.15
POL(c1(x1, x2)) = x1 + x2    2.87/1.15
POL(c2(x1)) = x1    2.87/1.15
POL(u) = [3]   
2.87/1.15
2.87/1.15

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2)) 2.87/1.15
+(+(z0, z1), z2) → +(z0, +(z1, z2)) 2.87/1.15
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2)) 2.87/1.15
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2)) 2.87/1.15
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(z1, z2))
S tuples:none
K tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2)) 2.87/1.15
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(z1, z2)) 2.87/1.15
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2

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2.87/1.15

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

The set S is empty
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2.87/1.15

(10) BOUNDS(O(1), O(1))

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2.87/1.15
3.40/1.30 EOF