YES(O(1), O(n^1)) 6.91/2.34 YES(O(1), O(n^1)) 6.91/2.36 6.91/2.36 6.91/2.36 6.91/2.36 6.91/2.36 6.91/2.36 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 6.91/2.36 6.91/2.36 6.91/2.36
6.91/2.36
6.91/2.36
6.91/2.36
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
6.91/2.36 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
6.91/2.36 
↳2 CdtProblem
6.91/2.36 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
6.91/2.36 
↳4 CdtProblem
6.91/2.36 
↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication)
6.91/2.36 
↳6 CdtProblem
6.91/2.36 
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
6.91/2.36 
↳8 CdtProblem
6.91/2.36 
↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
6.91/2.36 
↳10 CdtProblem
6.91/2.36 
↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
6.91/2.36 
↳12 CdtProblem
6.91/2.36 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.91/2.36 
↳14 CdtProblem
6.91/2.36 
↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
6.91/2.36 
↳16 CdtProblem
6.91/2.36 
↳17 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
6.91/2.36 
↳18 CdtProblem
6.91/2.36 
↳19 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
6.91/2.36 
↳20 CdtProblem
6.91/2.36 
↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.91/2.36 
↳22 CdtProblem
6.91/2.36 
↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
6.91/2.36 
↳24 CdtProblem
6.91/2.36 
↳25 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
6.91/2.36 
↳26 CdtProblem
6.91/2.36 
↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.91/2.36 
↳28 CdtProblem
6.91/2.36 
↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
6.91/2.36 
↳30 CdtProblem
6.91/2.36 
↳31 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
6.91/2.36 
↳32 CdtProblem
6.91/2.36 
↳33 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.91/2.36 
↳34 CdtProblem
6.91/2.36 
↳35 CdtKnowledgeProof (⇔)
6.91/2.36 
↳36 BOUNDS(O(1), O(1))
6.91/2.36
6.91/2.36 6.91/2.36
6.91/2.36
6.91/2.36

(0) Obligation:

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

nonZero(0) → false 6.91/2.36
nonZero(s(x)) → true 6.91/2.36
p(0) → 0 6.91/2.36
p(s(x)) → x 6.91/2.36
id_inc(x) → x 6.91/2.36
id_inc(x) → s(x) 6.91/2.36
random(x) → rand(x, 0) 6.91/2.36
rand(x, y) → if(nonZero(x), x, y) 6.91/2.36
if(false, x, y) → y 6.91/2.36
if(true, x, y) → rand(p(x), id_inc(y))

Rewrite Strategy: INNERMOST
6.91/2.36
6.91/2.36

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

Converted CpxTRS to CDT
6.91/2.36
6.91/2.36

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 6.91/2.36
nonZero(s(z0)) → true 6.91/2.36
p(0) → 0 6.91/2.36
p(s(z0)) → z0 6.91/2.36
id_inc(z0) → z0 6.91/2.36
id_inc(z0) → s(z0) 6.91/2.36
random(z0) → rand(z0, 0) 7.28/2.40
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.40
if(false, z0, z1) → z1 7.28/2.40
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

RANDOM(z0) → c6(RAND(z0, 0)) 7.28/2.40
RAND(z0, z1) → c7(IF(nonZero(z0), z0, z1), NONZERO(z0)) 7.28/2.40
IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)), P(z0), ID_INC(z1))
S tuples:

RANDOM(z0) → c6(RAND(z0, 0)) 7.28/2.40
RAND(z0, z1) → c7(IF(nonZero(z0), z0, z1), NONZERO(z0)) 7.28/2.40
IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)), P(z0), ID_INC(z1))
K tuples:none
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RANDOM, RAND, IF

Compound Symbols:

c6, c7, c9

7.28/2.40
7.28/2.40

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

Removed 3 trailing tuple parts
7.28/2.40
7.28/2.40

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.40
nonZero(s(z0)) → true 7.28/2.40
p(0) → 0 7.28/2.40
p(s(z0)) → z0 7.28/2.40
id_inc(z0) → z0 7.28/2.40
id_inc(z0) → s(z0) 7.28/2.40
random(z0) → rand(z0, 0) 7.28/2.40
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.40
if(false, z0, z1) → z1 7.28/2.40
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

RANDOM(z0) → c6(RAND(z0, 0)) 7.28/2.40
RAND(z0, z1) → c7(IF(nonZero(z0), z0, z1)) 7.28/2.40
IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)))
S tuples:

RANDOM(z0) → c6(RAND(z0, 0)) 7.28/2.40
RAND(z0, z1) → c7(IF(nonZero(z0), z0, z1)) 7.28/2.40
IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)))
K tuples:none
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RANDOM, RAND, IF

Compound Symbols:

c6, c7, c9

7.28/2.40
7.28/2.40

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

RANDOM(z0) → c6(RAND(z0, 0))
7.28/2.40
7.28/2.40

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.40
nonZero(s(z0)) → true 7.28/2.40
p(0) → 0 7.28/2.40
p(s(z0)) → z0 7.28/2.40
id_inc(z0) → z0 7.28/2.40
id_inc(z0) → s(z0) 7.28/2.40
random(z0) → rand(z0, 0) 7.28/2.40
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.40
if(false, z0, z1) → z1 7.28/2.40
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

RAND(z0, z1) → c7(IF(nonZero(z0), z0, z1)) 7.28/2.40
IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)))
S tuples:

RAND(z0, z1) → c7(IF(nonZero(z0), z0, z1)) 7.28/2.40
IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)))
K tuples:none
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c9

7.28/2.40
7.28/2.40

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

Use narrowing to replace RAND(z0, z1) → c7(IF(nonZero(z0), z0, z1)) by

RAND(0, x1) → c7(IF(false, 0, x1)) 7.28/2.40
RAND(s(z0), x1) → c7(IF(true, s(z0), x1))
7.28/2.40
7.28/2.40

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.40
nonZero(s(z0)) → true 7.28/2.40
p(0) → 0 7.28/2.40
p(s(z0)) → z0 7.28/2.40
id_inc(z0) → z0 7.28/2.40
id_inc(z0) → s(z0) 7.28/2.40
random(z0) → rand(z0, 0) 7.28/2.40
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.40
if(false, z0, z1) → z1 7.28/2.40
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1))) 7.28/2.40
RAND(0, x1) → c7(IF(false, 0, x1)) 7.28/2.40
RAND(s(z0), x1) → c7(IF(true, s(z0), x1))
S tuples:

IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1))) 7.28/2.40
RAND(0, x1) → c7(IF(false, 0, x1)) 7.28/2.40
RAND(s(z0), x1) → c7(IF(true, s(z0), x1))
K tuples:none
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

IF, RAND

Compound Symbols:

c9, c7

7.28/2.40
7.28/2.40

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

Removed 1 trailing tuple parts
7.28/2.40
7.28/2.40

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.40
nonZero(s(z0)) → true 7.28/2.40
p(0) → 0 7.28/2.40
p(s(z0)) → z0 7.28/2.40
id_inc(z0) → z0 7.28/2.40
id_inc(z0) → s(z0) 7.28/2.40
random(z0) → rand(z0, 0) 7.28/2.40
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.40
if(false, z0, z1) → z1 7.28/2.40
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1))) 7.28/2.40
RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.40
RAND(0, x1) → c7
S tuples:

IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1))) 7.28/2.40
RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.40
RAND(0, x1) → c7
K tuples:none
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

IF, RAND

Compound Symbols:

c9, c7, c7

7.28/2.40
7.28/2.40

(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

RAND(0, x1) → c7
7.28/2.40
7.28/2.40

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.40
nonZero(s(z0)) → true 7.28/2.40
p(0) → 0 7.28/2.40
p(s(z0)) → z0 7.28/2.40
id_inc(z0) → z0 7.28/2.40
id_inc(z0) → s(z0) 7.28/2.40
random(z0) → rand(z0, 0) 7.28/2.40
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.40
if(false, z0, z1) → z1 7.28/2.40
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1))) 7.28/2.42
RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7
S tuples:

IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1))) 7.28/2.42
RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7
K tuples:none
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

IF, RAND

Compound Symbols:

c9, c7, c7

7.28/2.42
7.28/2.42

(13) 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.

RAND(0, x1) → c7
We considered the (Usable) Rules:

p(0) → 0 7.28/2.42
p(s(z0)) → z0 7.28/2.42
id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0)
And the Tuples:

IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1))) 7.28/2.42
RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7
The order we found is given by the following interpretation:
Polynomial interpretation : 7.28/2.42

POL(0) = 0    7.28/2.42
POL(IF(x1, x2, x3)) = [1] + x1    7.28/2.42
POL(RAND(x1, x2)) = [1]    7.28/2.42
POL(c7) = 0    7.28/2.42
POL(c7(x1)) = x1    7.28/2.42
POL(c9(x1)) = x1    7.28/2.42
POL(id_inc(x1)) = [2] + [2]x1    7.28/2.42
POL(p(x1)) = 0    7.28/2.42
POL(s(x1)) = 0    7.28/2.42
POL(true) = 0   
7.28/2.42
7.28/2.42

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.42
nonZero(s(z0)) → true 7.28/2.42
p(0) → 0 7.28/2.42
p(s(z0)) → z0 7.28/2.42
id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0) 7.28/2.42
random(z0) → rand(z0, 0) 7.28/2.42
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.42
if(false, z0, z1) → z1 7.28/2.42
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1))) 7.28/2.42
RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7
S tuples:

IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1))) 7.28/2.42
RAND(s(z0), x1) → c7(IF(true, s(z0), x1))
K tuples:

RAND(0, x1) → c7
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

IF, RAND

Compound Symbols:

c9, c7, c7

7.28/2.42
7.28/2.42

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

Use narrowing to replace IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1))) by

IF(true, x0, z0) → c9(RAND(p(x0), z0)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)))
7.28/2.42
7.28/2.42

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.42
nonZero(s(z0)) → true 7.28/2.42
p(0) → 0 7.28/2.42
p(s(z0)) → z0 7.28/2.42
id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0) 7.28/2.42
random(z0) → rand(z0, 0) 7.28/2.42
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.42
if(false, z0, z1) → z1 7.28/2.42
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), z0)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)))
S tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), z0)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)))
K tuples:

RAND(0, x1) → c7
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c7, c9

7.28/2.42
7.28/2.42

(17) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1)))
7.28/2.42
7.28/2.42

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.42
nonZero(s(z0)) → true 7.28/2.42
p(0) → 0 7.28/2.42
p(s(z0)) → z0 7.28/2.42
id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0) 7.28/2.42
random(z0) → rand(z0, 0) 7.28/2.42
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.42
if(false, z0, z1) → z1 7.28/2.42
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), z0)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)))
S tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), z0)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)))
K tuples:

RAND(0, x1) → c7
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c7, c9

7.28/2.42
7.28/2.42

(19) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

The following tuples could be moved from S to K by knowledge propagation:

IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
RAND(0, x1) → c7
7.28/2.42
7.28/2.42

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.42
nonZero(s(z0)) → true 7.28/2.42
p(0) → 0 7.28/2.42
p(s(z0)) → z0 7.28/2.42
id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0) 7.28/2.42
random(z0) → rand(z0, 0) 7.28/2.42
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.42
if(false, z0, z1) → z1 7.28/2.42
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), z0)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)))
S tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), z0)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)))
K tuples:

RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1)))
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c7, c9

7.28/2.42
7.28/2.42

(21) 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.

IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)))
We considered the (Usable) Rules:

id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0) 7.28/2.42
p(0) → 0 7.28/2.42
p(s(z0)) → z0
And the Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), z0)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 7.28/2.42

POL(0) = 0    7.28/2.42
POL(IF(x1, x2, x3)) = [2]x2    7.28/2.42
POL(RAND(x1, x2)) = [2]x1    7.28/2.42
POL(c7) = 0    7.28/2.42
POL(c7(x1)) = x1    7.28/2.42
POL(c9(x1)) = x1    7.28/2.42
POL(id_inc(x1)) = x1    7.28/2.42
POL(p(x1)) = x1    7.28/2.42
POL(s(x1)) = [2] + x1    7.28/2.42
POL(true) = [1]   
7.28/2.42
7.28/2.42

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.42
nonZero(s(z0)) → true 7.28/2.42
p(0) → 0 7.28/2.42
p(s(z0)) → z0 7.28/2.42
id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0) 7.28/2.42
random(z0) → rand(z0, 0) 7.28/2.42
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.42
if(false, z0, z1) → z1 7.28/2.42
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), z0)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)))
S tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), z0)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0)))
K tuples:

RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)))
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c7, c9

7.28/2.42
7.28/2.42

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

Use narrowing to replace IF(true, x0, z0) → c9(RAND(p(x0), z0)) by

IF(true, 0, x1) → c9(RAND(0, x1)) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1))
7.28/2.42
7.28/2.42

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.42
nonZero(s(z0)) → true 7.28/2.42
p(0) → 0 7.28/2.42
p(s(z0)) → z0 7.28/2.42
id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0) 7.28/2.42
random(z0) → rand(z0, 0) 7.28/2.42
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.42
if(false, z0, z1) → z1 7.28/2.42
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1))) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, x1)) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1))
S tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, x1)) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1))
K tuples:

RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)))
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c7, c9

7.28/2.42
7.28/2.42

(25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, x1)) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1)))
7.28/2.42
7.28/2.42

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.42
nonZero(s(z0)) → true 7.28/2.42
p(0) → 0 7.28/2.42
p(s(z0)) → z0 7.28/2.42
id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0) 7.28/2.42
random(z0) → rand(z0, 0) 7.28/2.42
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.42
if(false, z0, z1) → z1 7.28/2.42
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1))
S tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1))
K tuples:

RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)))
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c7, c9

7.28/2.42
7.28/2.42

(27) 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.

IF(true, s(z0), x1) → c9(RAND(z0, x1))
We considered the (Usable) Rules:

id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0) 7.28/2.42
p(0) → 0 7.28/2.42
p(s(z0)) → z0
And the Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1))
The order we found is given by the following interpretation:
Polynomial interpretation : 7.28/2.42

POL(0) = 0    7.28/2.42
POL(IF(x1, x2, x3)) = [4]x2    7.28/2.42
POL(RAND(x1, x2)) = [4]x1    7.28/2.42
POL(c7) = 0    7.28/2.42
POL(c7(x1)) = x1    7.28/2.42
POL(c9(x1)) = x1    7.28/2.42
POL(id_inc(x1)) = [4] + x1    7.28/2.42
POL(p(x1)) = x1    7.28/2.42
POL(s(x1)) = [2] + x1    7.28/2.42
POL(true) = [1]   
7.28/2.42
7.28/2.42

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.42
nonZero(s(z0)) → true 7.28/2.42
p(0) → 0 7.28/2.42
p(s(z0)) → z0 7.28/2.42
id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0) 7.28/2.42
random(z0) → rand(z0, 0) 7.28/2.42
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.42
if(false, z0, z1) → z1 7.28/2.42
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1))
S tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
IF(true, x0, z0) → c9(RAND(p(x0), s(z0)))
K tuples:

RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1))
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c7, c9

7.28/2.42
7.28/2.42

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

Use narrowing to replace IF(true, x0, z0) → c9(RAND(p(x0), s(z0))) by

IF(true, 0, x1) → c9(RAND(0, s(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)))
7.28/2.42
7.28/2.42

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.42
nonZero(s(z0)) → true 7.28/2.42
p(0) → 0 7.28/2.42
p(s(z0)) → z0 7.28/2.42
id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0) 7.28/2.42
random(z0) → rand(z0, 0) 7.28/2.42
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.42
if(false, z0, z1) → z1 7.28/2.42
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1)) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, s(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)))
S tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, s(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)))
K tuples:

RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1))
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c7, c9

7.28/2.42
7.28/2.42

(31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

IF(true, 0, x1) → c9(RAND(0, s(x1))) 7.28/2.42
RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1)))
7.28/2.42
7.28/2.42

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.42
nonZero(s(z0)) → true 7.28/2.42
p(0) → 0 7.28/2.42
p(s(z0)) → z0 7.28/2.42
id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0) 7.28/2.42
random(z0) → rand(z0, 0) 7.28/2.42
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.42
if(false, z0, z1) → z1 7.28/2.42
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1)) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)))
S tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)))
K tuples:

RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1))
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c7, c9

7.28/2.42
7.28/2.42

(33) 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.

IF(true, s(z0), x1) → c9(RAND(z0, s(x1)))
We considered the (Usable) Rules:

id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0)
And the Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1)) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 7.28/2.42

POL(0) = 0    7.28/2.42
POL(IF(x1, x2, x3)) = [2]x2    7.28/2.42
POL(RAND(x1, x2)) = [2]x1    7.28/2.42
POL(c7) = 0    7.28/2.42
POL(c7(x1)) = x1    7.28/2.42
POL(c9(x1)) = x1    7.28/2.42
POL(id_inc(x1)) = x1    7.28/2.42
POL(s(x1)) = [1] + x1    7.28/2.42
POL(true) = [1]   
7.28/2.42
7.28/2.42

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false 7.28/2.42
nonZero(s(z0)) → true 7.28/2.42
p(0) → 0 7.28/2.42
p(s(z0)) → z0 7.28/2.42
id_inc(z0) → z0 7.28/2.42
id_inc(z0) → s(z0) 7.28/2.42
random(z0) → rand(z0, 0) 7.28/2.42
rand(z0, z1) → if(nonZero(z0), z0, z1) 7.28/2.42
if(false, z0, z1) → z1 7.28/2.42
if(true, z0, z1) → rand(p(z0), id_inc(z1))
Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1)) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)))
S tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1))
K tuples:

RAND(0, x1) → c7 7.28/2.42
IF(true, 0, x1) → c9(RAND(0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1)) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)))
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c7, c9

7.28/2.42
7.28/2.42

(35) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1)) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1))) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, x1)) 7.28/2.42
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)))
Now S is empty
7.28/2.42
7.28/2.42

(36) BOUNDS(O(1), O(1))

7.28/2.42
7.28/2.42
7.28/2.46 EOF