YES(O(1), O(n^2)) 6.01/2.25 YES(O(1), O(n^2)) 6.01/2.27 6.01/2.27 6.01/2.27 6.01/2.27 6.01/2.27 6.01/2.27 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 6.01/2.27 6.01/2.27 6.01/2.27
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
6.01/2.27 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
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↳2 CdtProblem
6.01/2.27 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
6.01/2.27 
↳4 CdtProblem
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↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
6.01/2.27 
↳6 CdtProblem
6.01/2.27 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.01/2.27 
↳8 CdtProblem
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↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
6.01/2.27 
↳10 CdtProblem
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↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
6.01/2.27 
↳12 CdtProblem
6.01/2.27 
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳14 BOUNDS(O(1), O(1))
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6.01/2.27

(0) Obligation:

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

leq(0, y) → true 6.01/2.27
leq(s(x), 0) → false 6.01/2.27
leq(s(x), s(y)) → leq(x, y) 6.01/2.27
if(true, x, y) → x 6.01/2.27
if(false, x, y) → y 6.01/2.27
-(x, 0) → x 6.01/2.27
-(s(x), s(y)) → -(x, y) 6.01/2.27
mod(0, y) → 0 6.01/2.27
mod(s(x), 0) → 0 6.01/2.27
mod(s(x), s(y)) → if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Rewrite Strategy: INNERMOST
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6.01/2.27

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

Converted CpxTRS to CDT
6.01/2.27
6.01/2.27

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

leq(0, z0) → true 6.01/2.27
leq(s(z0), 0) → false 6.01/2.27
leq(s(z0), s(z1)) → leq(z0, z1) 6.01/2.27
if(true, z0, z1) → z0 6.01/2.27
if(false, z0, z1) → z1 6.01/2.27
-(z0, 0) → z0 6.01/2.27
-(s(z0), s(z1)) → -(z0, z1) 6.01/2.27
mod(0, z0) → 0 6.01/2.27
mod(s(z0), 0) → 0 6.01/2.27
mod(s(z0), s(z1)) → if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1)) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1)) 6.01/2.29
MOD(s(z0), s(z1)) → c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1)) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1)) 6.01/2.29
MOD(s(z0), s(z1)) → c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

leq, if, -, mod

Defined Pair Symbols:

LEQ, -', MOD

Compound Symbols:

c2, c6, c9

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

Removed 1 trailing tuple parts
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6.01/2.29

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

leq(0, z0) → true 6.01/2.29
leq(s(z0), 0) → false 6.01/2.29
leq(s(z0), s(z1)) → leq(z0, z1) 6.01/2.29
if(true, z0, z1) → z0 6.01/2.29
if(false, z0, z1) → z1 6.01/2.29
-(z0, 0) → z0 6.01/2.29
-(s(z0), s(z1)) → -(z0, z1) 6.01/2.29
mod(0, z0) → 0 6.01/2.29
mod(s(z0), 0) → 0 6.01/2.29
mod(s(z0), s(z1)) → if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1)) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1)) 6.01/2.29
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1)) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1)) 6.01/2.29
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

leq, if, -, mod

Defined Pair Symbols:

LEQ, -', MOD

Compound Symbols:

c2, c6, c9

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

Use narrowing to replace MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1))) by

MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
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6.01/2.29

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

leq(0, z0) → true 6.01/2.29
leq(s(z0), 0) → false 6.01/2.29
leq(s(z0), s(z1)) → leq(z0, z1) 6.01/2.29
if(true, z0, z1) → z0 6.01/2.29
if(false, z0, z1) → z1 6.01/2.29
-(z0, 0) → z0 6.01/2.29
-(s(z0), s(z1)) → -(z0, z1) 6.01/2.29
mod(0, z0) → 0 6.01/2.29
mod(s(z0), 0) → 0 6.01/2.29
mod(s(z0), s(z1)) → if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1)) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1)) 6.01/2.29
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
S tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1)) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1)) 6.01/2.29
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

leq, if, -, mod

Defined Pair Symbols:

LEQ, -', MOD

Compound Symbols:

c2, c6, c9

6.01/2.29
6.01/2.29

(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.

MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
We considered the (Usable) Rules:

-(z0, 0) → z0 6.01/2.29
-(s(z0), s(z1)) → -(z0, z1)
And the Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1)) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1)) 6.01/2.29
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 6.01/2.29

POL(-(x1, x2)) = [2] + x1    6.01/2.29
POL(-'(x1, x2)) = [3]    6.01/2.29
POL(0) = 0    6.01/2.29
POL(LEQ(x1, x2)) = 0    6.01/2.29
POL(MOD(x1, x2)) = [2]x1    6.01/2.29
POL(c2(x1)) = x1    6.01/2.29
POL(c6(x1)) = x1    6.01/2.29
POL(c9(x1, x2, x3)) = x1 + x2 + x3    6.01/2.29
POL(s(x1)) = [5] + x1   
6.01/2.29
6.01/2.29

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

leq(0, z0) → true 6.01/2.29
leq(s(z0), 0) → false 6.01/2.29
leq(s(z0), s(z1)) → leq(z0, z1) 6.01/2.29
if(true, z0, z1) → z0 6.01/2.29
if(false, z0, z1) → z1 6.01/2.29
-(z0, 0) → z0 6.01/2.29
-(s(z0), s(z1)) → -(z0, z1) 6.01/2.29
mod(0, z0) → 0 6.01/2.29
mod(s(z0), 0) → 0 6.01/2.29
mod(s(z0), s(z1)) → if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1)) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1)) 6.01/2.29
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
S tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1)) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1))
K tuples:

MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
Defined Rule Symbols:

leq, if, -, mod

Defined Pair Symbols:

LEQ, -', MOD

Compound Symbols:

c2, c6, c9

6.01/2.29
6.01/2.29

(9) 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.

-'(s(z0), s(z1)) → c6(-'(z0, z1))
We considered the (Usable) Rules:

-(z0, 0) → z0 6.01/2.29
-(s(z0), s(z1)) → -(z0, z1)
And the Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1)) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1)) 6.01/2.29
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 6.01/2.29

POL(-(x1, x2)) = x1    6.01/2.29
POL(-'(x1, x2)) = x1    6.01/2.29
POL(0) = [3]    6.01/2.29
POL(LEQ(x1, x2)) = 0    6.01/2.29
POL(MOD(x1, x2)) = x12    6.01/2.29
POL(c2(x1)) = x1    6.01/2.29
POL(c6(x1)) = x1    6.01/2.29
POL(c9(x1, x2, x3)) = x1 + x2 + x3    6.01/2.29
POL(s(x1)) = [1] + x1   
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6.01/2.29

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

leq(0, z0) → true 6.01/2.29
leq(s(z0), 0) → false 6.01/2.29
leq(s(z0), s(z1)) → leq(z0, z1) 6.01/2.29
if(true, z0, z1) → z0 6.01/2.29
if(false, z0, z1) → z1 6.01/2.29
-(z0, 0) → z0 6.01/2.29
-(s(z0), s(z1)) → -(z0, z1) 6.01/2.29
mod(0, z0) → 0 6.01/2.29
mod(s(z0), 0) → 0 6.01/2.29
mod(s(z0), s(z1)) → if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1)) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1)) 6.01/2.29
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
S tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
K tuples:

MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1))) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1))
Defined Rule Symbols:

leq, if, -, mod

Defined Pair Symbols:

LEQ, -', MOD

Compound Symbols:

c2, c6, c9

6.01/2.29
6.01/2.29

(11) 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.

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
We considered the (Usable) Rules:

-(z0, 0) → z0 6.01/2.29
-(s(z0), s(z1)) → -(z0, z1)
And the Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1)) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1)) 6.01/2.29
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 6.01/2.29

POL(-(x1, x2)) = x1    6.01/2.29
POL(-'(x1, x2)) = 0    6.01/2.29
POL(0) = [3]    6.01/2.29
POL(LEQ(x1, x2)) = x2    6.01/2.29
POL(MOD(x1, x2)) = x12    6.01/2.29
POL(c2(x1)) = x1    6.01/2.29
POL(c6(x1)) = x1    6.01/2.29
POL(c9(x1, x2, x3)) = x1 + x2 + x3    6.01/2.29
POL(s(x1)) = [1] + x1   
6.01/2.29
6.01/2.29

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

leq(0, z0) → true 6.01/2.29
leq(s(z0), 0) → false 6.01/2.29
leq(s(z0), s(z1)) → leq(z0, z1) 6.01/2.29
if(true, z0, z1) → z0 6.01/2.29
if(false, z0, z1) → z1 6.01/2.29
-(z0, 0) → z0 6.01/2.29
-(s(z0), s(z1)) → -(z0, z1) 6.01/2.29
mod(0, z0) → 0 6.01/2.29
mod(s(z0), 0) → 0 6.01/2.29
mod(s(z0), s(z1)) → if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1)) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1)) 6.01/2.29
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
S tuples:none
K tuples:

MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1))) 6.01/2.29
-'(s(z0), s(z1)) → c6(-'(z0, z1)) 6.01/2.29
LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
Defined Rule Symbols:

leq, if, -, mod

Defined Pair Symbols:

LEQ, -', MOD

Compound Symbols:

c2, c6, c9

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6.01/2.29

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

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

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6.34/2.34 EOF