YES(O(1), O(n^1)) 0.00/0.82 YES(O(1), O(n^1)) 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.84 0.00/0.84 0.00/0.84
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0.00/0.84
0.00/0.84
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))
0.00/0.84 
↳2 CdtProblem
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↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
0.00/0.84 
↳4 CdtProblem
0.00/0.84 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.84 
↳6 CdtProblem
0.00/0.84 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.84 
↳8 CdtProblem
0.00/0.84 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.84 
↳10 CdtProblem
0.00/0.84 
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID))
0.00/0.84 
↳12 BOUNDS(O(1), O(1))
0.00/0.84
0.00/0.84 0.00/0.84
0.00/0.84
0.00/0.84

(0) Obligation:

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

not(true) → false 0.00/0.84
not(false) → true 0.00/0.84
odd(0) → false 0.00/0.84
odd(s(x)) → not(odd(x)) 0.00/0.84
+(x, 0) → x 0.00/0.84
+(x, s(y)) → s(+(x, y)) 0.00/0.84
+(s(x), y) → s(+(x, y))

Rewrite Strategy: INNERMOST
0.00/0.84
0.00/0.84

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

Converted CpxTRS to CDT
0.00/0.84
0.00/0.84

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

not(true) → false 0.00/0.84
not(false) → true 0.00/0.84
odd(0) → false 0.00/0.84
odd(s(z0)) → not(odd(z0)) 0.00/0.84
+(z0, 0) → z0 0.00/0.84
+(z0, s(z1)) → s(+(z0, z1)) 0.00/0.84
+(s(z0), z1) → s(+(z0, z1))
Tuples:

ODD(s(z0)) → c3(NOT(odd(z0)), ODD(z0)) 0.00/0.84
+'(z0, s(z1)) → c5(+'(z0, z1)) 0.00/0.84
+'(s(z0), z1) → c6(+'(z0, z1))
S tuples:

ODD(s(z0)) → c3(NOT(odd(z0)), ODD(z0)) 0.00/0.84
+'(z0, s(z1)) → c5(+'(z0, z1)) 0.00/0.84
+'(s(z0), z1) → c6(+'(z0, z1))
K tuples:none
Defined Rule Symbols:

not, odd, +

Defined Pair Symbols:

ODD, +'

Compound Symbols:

c3, c5, c6

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0.00/0.84

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

Removed 1 trailing tuple parts
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0.00/0.84

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

not(true) → false 0.00/0.84
not(false) → true 0.00/0.84
odd(0) → false 0.00/0.84
odd(s(z0)) → not(odd(z0)) 0.00/0.84
+(z0, 0) → z0 0.00/0.84
+(z0, s(z1)) → s(+(z0, z1)) 0.00/0.84
+(s(z0), z1) → s(+(z0, z1))
Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1)) 0.00/0.84
+'(s(z0), z1) → c6(+'(z0, z1)) 0.00/0.84
ODD(s(z0)) → c3(ODD(z0))
S tuples:

+'(z0, s(z1)) → c5(+'(z0, z1)) 0.00/0.84
+'(s(z0), z1) → c6(+'(z0, z1)) 0.00/0.84
ODD(s(z0)) → c3(ODD(z0))
K tuples:none
Defined Rule Symbols:

not, odd, +

Defined Pair Symbols:

+', ODD

Compound Symbols:

c5, c6, c3

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0.00/0.84

(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, s(z1)) → c5(+'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1)) 0.00/0.84
+'(s(z0), z1) → c6(+'(z0, z1)) 0.00/0.84
ODD(s(z0)) → c3(ODD(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.84

POL(+'(x1, x2)) = x2    0.00/0.84
POL(ODD(x1)) = 0    0.00/0.84
POL(c3(x1)) = x1    0.00/0.84
POL(c5(x1)) = x1    0.00/0.84
POL(c6(x1)) = x1    0.00/0.84
POL(s(x1)) = [1] + x1   
0.00/0.84
0.00/0.84

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

not(true) → false 0.00/0.84
not(false) → true 0.00/0.84
odd(0) → false 0.00/0.84
odd(s(z0)) → not(odd(z0)) 0.00/0.84
+(z0, 0) → z0 0.00/0.84
+(z0, s(z1)) → s(+(z0, z1)) 0.00/0.84
+(s(z0), z1) → s(+(z0, z1))
Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1)) 0.00/0.84
+'(s(z0), z1) → c6(+'(z0, z1)) 0.00/0.84
ODD(s(z0)) → c3(ODD(z0))
S tuples:

+'(s(z0), z1) → c6(+'(z0, z1)) 0.00/0.84
ODD(s(z0)) → c3(ODD(z0))
K tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
Defined Rule Symbols:

not, odd, +

Defined Pair Symbols:

+', ODD

Compound Symbols:

c5, c6, c3

0.00/0.84
0.00/0.84

(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.

+'(s(z0), z1) → c6(+'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1)) 0.00/0.84
+'(s(z0), z1) → c6(+'(z0, z1)) 0.00/0.84
ODD(s(z0)) → c3(ODD(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.84

POL(+'(x1, x2)) = x1 + [5]x2    0.00/0.84
POL(ODD(x1)) = 0    0.00/0.84
POL(c3(x1)) = x1    0.00/0.84
POL(c5(x1)) = x1    0.00/0.84
POL(c6(x1)) = x1    0.00/0.84
POL(s(x1)) = [1] + x1   
0.00/0.84
0.00/0.84

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

not(true) → false 0.00/0.84
not(false) → true 0.00/0.84
odd(0) → false 0.00/0.84
odd(s(z0)) → not(odd(z0)) 0.00/0.84
+(z0, 0) → z0 0.00/0.84
+(z0, s(z1)) → s(+(z0, z1)) 0.00/0.84
+(s(z0), z1) → s(+(z0, z1))
Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1)) 0.00/0.84
+'(s(z0), z1) → c6(+'(z0, z1)) 0.00/0.84
ODD(s(z0)) → c3(ODD(z0))
S tuples:

ODD(s(z0)) → c3(ODD(z0))
K tuples:

+'(z0, s(z1)) → c5(+'(z0, z1)) 0.00/0.84
+'(s(z0), z1) → c6(+'(z0, z1))
Defined Rule Symbols:

not, odd, +

Defined Pair Symbols:

+', ODD

Compound Symbols:

c5, c6, c3

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0.00/0.84

(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.

ODD(s(z0)) → c3(ODD(z0))
We considered the (Usable) Rules:none
And the Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1)) 0.00/0.85
+'(s(z0), z1) → c6(+'(z0, z1)) 0.00/0.85
ODD(s(z0)) → c3(ODD(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.85

POL(+'(x1, x2)) = [5]x1 + [5]x2    0.00/0.85
POL(ODD(x1)) = [3]x1    0.00/0.85
POL(c3(x1)) = x1    0.00/0.85
POL(c5(x1)) = x1    0.00/0.85
POL(c6(x1)) = x1    0.00/0.85
POL(s(x1)) = [1] + x1   
0.00/0.85
0.00/0.85

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

not(true) → false 0.00/0.85
not(false) → true 0.00/0.85
odd(0) → false 0.00/0.85
odd(s(z0)) → not(odd(z0)) 0.00/0.85
+(z0, 0) → z0 0.00/0.85
+(z0, s(z1)) → s(+(z0, z1)) 0.00/0.85
+(s(z0), z1) → s(+(z0, z1))
Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1)) 0.00/0.85
+'(s(z0), z1) → c6(+'(z0, z1)) 0.00/0.85
ODD(s(z0)) → c3(ODD(z0))
S tuples:none
K tuples:

+'(z0, s(z1)) → c5(+'(z0, z1)) 0.00/0.85
+'(s(z0), z1) → c6(+'(z0, z1)) 0.00/0.85
ODD(s(z0)) → c3(ODD(z0))
Defined Rule Symbols:

not, odd, +

Defined Pair Symbols:

+', ODD

Compound Symbols:

c5, c6, c3

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0.00/0.85

(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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0.00/0.85
0.00/0.89 EOF