YES(O(1), O(n^2)) 13.83/7.85 YES(O(1), O(n^2)) 13.83/7.87 13.83/7.87 13.83/7.87 13.83/7.87 13.83/7.87 13.83/7.87 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 13.83/7.87 13.83/7.87 13.83/7.87
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13.83/7.87
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
13.83/7.87 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
13.83/7.87 
↳2 CdtProblem
13.83/7.87 
↳3 CdtUnreachableProof (⇔)
13.83/7.87 
↳4 CdtProblem
13.83/7.87 
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
13.83/7.87 
↳6 CdtProblem
13.83/7.87 
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
13.83/7.87 
↳8 CdtProblem
13.83/7.87 
↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
13.83/7.87 
↳10 CdtProblem
13.83/7.87 
↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
13.83/7.87 
↳12 CdtProblem
13.83/7.87 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
13.83/7.87 
↳14 CdtProblem
13.83/7.87 
↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
13.83/7.87 
↳16 CdtProblem
13.83/7.87 
↳17 CdtLeafRemovalProof (ComplexityIfPolyImplication)
13.83/7.87 
↳18 CdtProblem
13.83/7.87 
↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
13.83/7.87 
↳20 CdtProblem
13.83/7.87 
↳21 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
13.83/7.87 
↳22 CdtProblem
13.83/7.87 
↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
13.83/7.87 
↳24 CdtProblem
13.83/7.87 
↳25 SIsEmptyProof (BOTH BOUNDS(ID, ID))
13.83/7.87 
↳26 BOUNDS(O(1), O(1))
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13.83/7.87

(0) Obligation:

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

p(0) → s(s(0)) 13.83/7.87
p(s(x)) → x 13.83/7.87
p(p(s(x))) → p(x) 13.83/7.87
le(p(s(x)), x) → le(x, x) 13.83/7.87
le(0, y) → true 13.83/7.87
le(s(x), 0) → false 13.83/7.87
le(s(x), s(y)) → le(x, y) 13.83/7.87
minus(x, y) → if(le(x, y), x, y) 13.83/7.87
if(true, x, y) → 0 13.83/7.87
if(false, x, y) → s(minus(p(x), y))

Rewrite Strategy: INNERMOST
13.83/7.87
13.83/7.87

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

Converted CpxTRS to CDT
13.83/7.87
13.83/7.87

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → s(s(0)) 13.83/7.87
p(s(z0)) → z0 13.83/7.87
p(p(s(z0))) → p(z0) 13.83/7.87
le(p(s(z0)), z0) → le(z0, z0) 13.83/7.87
le(0, z0) → true 13.83/7.87
le(s(z0), 0) → false 13.83/7.87
le(s(z0), s(z1)) → le(z0, z1) 13.83/7.87
minus(z0, z1) → if(le(z0, z1), z0, z1) 13.83/7.90
if(true, z0, z1) → 0 13.83/7.90
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

P(p(s(z0))) → c2(P(z0)) 13.83/7.90
LE(p(s(z0)), z0) → c3(LE(z0, z0)) 13.83/7.90
LE(s(z0), s(z1)) → c6(LE(z0, z1)) 13.83/7.90
MINUS(z0, z1) → c7(IF(le(z0, z1), z0, z1), LE(z0, z1)) 13.83/7.90
IF(false, z0, z1) → c9(MINUS(p(z0), z1), P(z0))
S tuples:

P(p(s(z0))) → c2(P(z0)) 13.83/7.90
LE(p(s(z0)), z0) → c3(LE(z0, z0)) 13.83/7.90
LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.90
MINUS(z0, z1) → c7(IF(le(z0, z1), z0, z1), LE(z0, z1)) 14.19/7.90
IF(false, z0, z1) → c9(MINUS(p(z0), z1), P(z0))
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

P, LE, MINUS, IF

Compound Symbols:

c2, c3, c6, c7, c9

14.19/7.90
14.19/7.90

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

P(p(s(z0))) → c2(P(z0)) 14.19/7.90
LE(p(s(z0)), z0) → c3(LE(z0, z0))
14.19/7.90
14.19/7.90

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → s(s(0)) 14.19/7.90
p(s(z0)) → z0 14.19/7.90
p(p(s(z0))) → p(z0) 14.19/7.90
le(p(s(z0)), z0) → le(z0, z0) 14.19/7.90
le(0, z0) → true 14.19/7.90
le(s(z0), 0) → false 14.19/7.90
le(s(z0), s(z1)) → le(z0, z1) 14.19/7.90
minus(z0, z1) → if(le(z0, z1), z0, z1) 14.19/7.90
if(true, z0, z1) → 0 14.19/7.90
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.90
MINUS(z0, z1) → c7(IF(le(z0, z1), z0, z1), LE(z0, z1)) 14.19/7.90
IF(false, z0, z1) → c9(MINUS(p(z0), z1), P(z0))
S tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.90
MINUS(z0, z1) → c7(IF(le(z0, z1), z0, z1), LE(z0, z1)) 14.19/7.90
IF(false, z0, z1) → c9(MINUS(p(z0), z1), P(z0))
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c6, c7, c9

14.19/7.90
14.19/7.90

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

Removed 1 trailing tuple parts
14.19/7.90
14.19/7.90

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → s(s(0)) 14.19/7.90
p(s(z0)) → z0 14.19/7.90
p(p(s(z0))) → p(z0) 14.19/7.90
le(p(s(z0)), z0) → le(z0, z0) 14.19/7.90
le(0, z0) → true 14.19/7.90
le(s(z0), 0) → false 14.19/7.90
le(s(z0), s(z1)) → le(z0, z1) 14.19/7.90
minus(z0, z1) → if(le(z0, z1), z0, z1) 14.19/7.90
if(true, z0, z1) → 0 14.19/7.90
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.90
MINUS(z0, z1) → c7(IF(le(z0, z1), z0, z1), LE(z0, z1)) 14.19/7.90
IF(false, z0, z1) → c9(MINUS(p(z0), z1))
S tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.90
MINUS(z0, z1) → c7(IF(le(z0, z1), z0, z1), LE(z0, z1)) 14.19/7.90
IF(false, z0, z1) → c9(MINUS(p(z0), z1))
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c6, c7, c9

14.19/7.90
14.19/7.90

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

Use narrowing to replace MINUS(z0, z1) → c7(IF(le(z0, z1), z0, z1), LE(z0, z1)) by

MINUS(0, z0) → c7(IF(true, 0, z0), LE(0, z0)) 14.19/7.90
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0), LE(s(z0), 0)) 14.19/7.90
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
14.19/7.90
14.19/7.90

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → s(s(0)) 14.19/7.90
p(s(z0)) → z0 14.19/7.90
p(p(s(z0))) → p(z0) 14.19/7.90
le(p(s(z0)), z0) → le(z0, z0) 14.19/7.90
le(0, z0) → true 14.19/7.90
le(s(z0), 0) → false 14.19/7.90
le(s(z0), s(z1)) → le(z0, z1) 14.19/7.90
minus(z0, z1) → if(le(z0, z1), z0, z1) 14.19/7.90
if(true, z0, z1) → 0 14.19/7.90
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.90
IF(false, z0, z1) → c9(MINUS(p(z0), z1)) 14.19/7.90
MINUS(0, z0) → c7(IF(true, 0, z0), LE(0, z0)) 14.19/7.90
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0), LE(s(z0), 0)) 14.19/7.90
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.90
IF(false, z0, z1) → c9(MINUS(p(z0), z1)) 14.19/7.90
MINUS(0, z0) → c7(IF(true, 0, z0), LE(0, z0)) 14.19/7.90
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0), LE(s(z0), 0)) 14.19/7.90
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, IF, MINUS

Compound Symbols:

c6, c9, c7

14.19/7.90
14.19/7.90

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

Removed 3 trailing tuple parts
14.19/7.90
14.19/7.90

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → s(s(0)) 14.19/7.90
p(s(z0)) → z0 14.19/7.90
p(p(s(z0))) → p(z0) 14.19/7.90
le(p(s(z0)), z0) → le(z0, z0) 14.19/7.90
le(0, z0) → true 14.19/7.90
le(s(z0), 0) → false 14.19/7.90
le(s(z0), s(z1)) → le(z0, z1) 14.19/7.90
minus(z0, z1) → if(le(z0, z1), z0, z1) 14.19/7.90
if(true, z0, z1) → 0 14.19/7.90
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.90
IF(false, z0, z1) → c9(MINUS(p(z0), z1)) 14.19/7.90
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.90
MINUS(0, z0) → c7 14.19/7.90
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0))
S tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.90
IF(false, z0, z1) → c9(MINUS(p(z0), z1)) 14.19/7.90
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.90
MINUS(0, z0) → c7 14.19/7.90
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0))
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, IF, MINUS

Compound Symbols:

c6, c9, c7, c7, c7

14.19/7.90
14.19/7.90

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

Removed 1 trailing nodes:

MINUS(0, z0) → c7
14.19/7.90
14.19/7.90

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → s(s(0)) 14.19/7.90
p(s(z0)) → z0 14.19/7.90
p(p(s(z0))) → p(z0) 14.19/7.90
le(p(s(z0)), z0) → le(z0, z0) 14.19/7.91
le(0, z0) → true 14.19/7.91
le(s(z0), 0) → false 14.19/7.91
le(s(z0), s(z1)) → le(z0, z1) 14.19/7.91
minus(z0, z1) → if(le(z0, z1), z0, z1) 14.19/7.91
if(true, z0, z1) → 0 14.19/7.91
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.91
IF(false, z0, z1) → c9(MINUS(p(z0), z1)) 14.19/7.91
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.91
MINUS(0, z0) → c7 14.19/7.91
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0))
S tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.91
IF(false, z0, z1) → c9(MINUS(p(z0), z1)) 14.19/7.91
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.91
MINUS(0, z0) → c7 14.19/7.91
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0))
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, IF, MINUS

Compound Symbols:

c6, c9, c7, c7, c7

14.19/7.91
14.19/7.91

(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.

MINUS(0, z0) → c7
We considered the (Usable) Rules:

le(0, z0) → true 14.19/7.91
le(s(z0), 0) → false 14.19/7.91
le(s(z0), s(z1)) → le(z0, z1) 14.19/7.91
p(0) → s(s(0)) 14.19/7.91
p(s(z0)) → z0
And the Tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.91
IF(false, z0, z1) → c9(MINUS(p(z0), z1)) 14.19/7.91
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.91
MINUS(0, z0) → c7 14.19/7.91
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0))
The order we found is given by the following interpretation:
Polynomial interpretation : 14.19/7.91

POL(0) = [4]    14.19/7.91
POL(IF(x1, x2, x3)) = [2]x2    14.19/7.91
POL(LE(x1, x2)) = 0    14.19/7.91
POL(MINUS(x1, x2)) = [2]x1    14.19/7.91
POL(c6(x1)) = x1    14.19/7.91
POL(c7) = 0    14.19/7.91
POL(c7(x1)) = x1    14.19/7.91
POL(c7(x1, x2)) = x1 + x2    14.19/7.91
POL(c9(x1)) = x1    14.19/7.91
POL(false) = 0    14.19/7.91
POL(le(x1, x2)) = 0    14.19/7.91
POL(p(x1)) = x1    14.19/7.91
POL(s(x1)) = x1    14.19/7.91
POL(true) = [3]   
14.19/7.91
14.19/7.91

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → s(s(0)) 14.19/7.91
p(s(z0)) → z0 14.19/7.91
p(p(s(z0))) → p(z0) 14.19/7.91
le(p(s(z0)), z0) → le(z0, z0) 14.19/7.91
le(0, z0) → true 14.19/7.91
le(s(z0), 0) → false 14.19/7.91
le(s(z0), s(z1)) → le(z0, z1) 14.19/7.91
minus(z0, z1) → if(le(z0, z1), z0, z1) 14.19/7.91
if(true, z0, z1) → 0 14.19/7.91
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.91
IF(false, z0, z1) → c9(MINUS(p(z0), z1)) 14.19/7.91
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.91
MINUS(0, z0) → c7 14.19/7.91
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0))
S tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.91
IF(false, z0, z1) → c9(MINUS(p(z0), z1)) 14.19/7.91
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.91
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0))
K tuples:

MINUS(0, z0) → c7
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, IF, MINUS

Compound Symbols:

c6, c9, c7, c7, c7

14.19/7.91
14.19/7.91

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

Use narrowing to replace IF(false, z0, z1) → c9(MINUS(p(z0), z1)) by

IF(false, 0, x1) → c9(MINUS(s(s(0)), x1)) 14.19/7.91
IF(false, s(z0), x1) → c9(MINUS(z0, x1))
14.19/7.91
14.19/7.91

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → s(s(0)) 14.19/7.91
p(s(z0)) → z0 14.19/7.91
p(p(s(z0))) → p(z0) 14.19/7.91
le(p(s(z0)), z0) → le(z0, z0) 14.19/7.91
le(0, z0) → true 14.19/7.91
le(s(z0), 0) → false 14.19/7.91
le(s(z0), s(z1)) → le(z0, z1) 14.19/7.91
minus(z0, z1) → if(le(z0, z1), z0, z1) 14.19/7.91
if(true, z0, z1) → 0 14.19/7.91
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.91
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.91
MINUS(0, z0) → c7 14.19/7.91
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0)) 14.19/7.91
IF(false, 0, x1) → c9(MINUS(s(s(0)), x1)) 14.19/7.91
IF(false, s(z0), x1) → c9(MINUS(z0, x1))
S tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.91
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.91
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0)) 14.19/7.91
IF(false, 0, x1) → c9(MINUS(s(s(0)), x1)) 14.19/7.91
IF(false, s(z0), x1) → c9(MINUS(z0, x1))
K tuples:

MINUS(0, z0) → c7
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c6, c7, c7, c7, c9

14.19/7.92
14.19/7.92

(17) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

IF(false, 0, x1) → c9(MINUS(s(s(0)), x1))
Removed 1 trailing nodes:

MINUS(0, z0) → c7
14.19/7.92
14.19/7.92

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → s(s(0)) 14.19/7.92
p(s(z0)) → z0 14.19/7.92
p(p(s(z0))) → p(z0) 14.19/7.92
le(p(s(z0)), z0) → le(z0, z0) 14.19/7.92
le(0, z0) → true 14.19/7.92
le(s(z0), 0) → false 14.19/7.92
le(s(z0), s(z1)) → le(z0, z1) 14.19/7.92
minus(z0, z1) → if(le(z0, z1), z0, z1) 14.19/7.92
if(true, z0, z1) → 0 14.19/7.92
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.92
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.92
MINUS(0, z0) → c7 14.19/7.92
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0)) 14.19/7.92
IF(false, s(z0), x1) → c9(MINUS(z0, x1))
S tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.92
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.92
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0)) 14.19/7.92
IF(false, s(z0), x1) → c9(MINUS(z0, x1))
K tuples:

MINUS(0, z0) → c7
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c6, c7, c7, c7, c9

14.19/7.92
14.19/7.92

(19) 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(false, s(z0), x1) → c9(MINUS(z0, x1))
We considered the (Usable) Rules:

le(0, z0) → true 14.19/7.92
le(s(z0), 0) → false 14.19/7.92
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.92
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.92
MINUS(0, z0) → c7 14.19/7.92
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0)) 14.19/7.92
IF(false, s(z0), x1) → c9(MINUS(z0, x1))
The order we found is given by the following interpretation:
Polynomial interpretation : 14.19/7.92

POL(0) = 0    14.19/7.92
POL(IF(x1, x2, x3)) = x2    14.19/7.92
POL(LE(x1, x2)) = 0    14.19/7.92
POL(MINUS(x1, x2)) = x1    14.19/7.92
POL(c6(x1)) = x1    14.19/7.92
POL(c7) = 0    14.19/7.92
POL(c7(x1)) = x1    14.19/7.92
POL(c7(x1, x2)) = x1 + x2    14.19/7.92
POL(c9(x1)) = x1    14.19/7.92
POL(false) = 0    14.19/7.92
POL(le(x1, x2)) = 0    14.19/7.92
POL(s(x1)) = [1] + x1    14.19/7.92
POL(true) = [3]   
14.19/7.92
14.19/7.92

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → s(s(0)) 14.19/7.92
p(s(z0)) → z0 14.19/7.92
p(p(s(z0))) → p(z0) 14.19/7.92
le(p(s(z0)), z0) → le(z0, z0) 14.19/7.92
le(0, z0) → true 14.19/7.92
le(s(z0), 0) → false 14.19/7.92
le(s(z0), s(z1)) → le(z0, z1) 14.19/7.92
minus(z0, z1) → if(le(z0, z1), z0, z1) 14.19/7.92
if(true, z0, z1) → 0 14.19/7.92
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.92
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.92
MINUS(0, z0) → c7 14.19/7.92
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0)) 14.19/7.92
IF(false, s(z0), x1) → c9(MINUS(z0, x1))
S tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.92
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.92
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0))
K tuples:

MINUS(0, z0) → c7 14.19/7.92
IF(false, s(z0), x1) → c9(MINUS(z0, x1))
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c6, c7, c7, c7, c9

14.19/7.92
14.19/7.92

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

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

MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.92
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0)) 14.19/7.92
IF(false, s(z0), x1) → c9(MINUS(z0, x1)) 14.19/7.92
IF(false, s(z0), x1) → c9(MINUS(z0, x1))
14.19/7.92
14.19/7.92

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → s(s(0)) 14.19/7.92
p(s(z0)) → z0 14.19/7.92
p(p(s(z0))) → p(z0) 14.19/7.92
le(p(s(z0)), z0) → le(z0, z0) 14.19/7.92
le(0, z0) → true 14.19/7.92
le(s(z0), 0) → false 14.19/7.92
le(s(z0), s(z1)) → le(z0, z1) 14.19/7.92
minus(z0, z1) → if(le(z0, z1), z0, z1) 14.19/7.92
if(true, z0, z1) → 0 14.19/7.92
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.92
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.92
MINUS(0, z0) → c7 14.19/7.92
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0)) 14.19/7.92
IF(false, s(z0), x1) → c9(MINUS(z0, x1))
S tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1))
K tuples:

MINUS(0, z0) → c7 14.19/7.92
IF(false, s(z0), x1) → c9(MINUS(z0, x1)) 14.19/7.92
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.92
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0))
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c6, c7, c7, c7, c9

14.19/7.92
14.19/7.92

(23) 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.

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

le(0, z0) → true 14.19/7.92
le(s(z0), 0) → false 14.19/7.92
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.92
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.92
MINUS(0, z0) → c7 14.19/7.92
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0)) 14.19/7.92
IF(false, s(z0), x1) → c9(MINUS(z0, x1))
The order we found is given by the following interpretation:
Polynomial interpretation : 14.19/7.92

POL(0) = 0    14.19/7.92
POL(IF(x1, x2, x3)) = [2] + x22    14.19/7.92
POL(LE(x1, x2)) = x1    14.19/7.92
POL(MINUS(x1, x2)) = [2] + [2]x1 + x12    14.19/7.92
POL(c6(x1)) = x1    14.19/7.92
POL(c7) = 0    14.19/7.92
POL(c7(x1)) = x1    14.19/7.92
POL(c7(x1, x2)) = x1 + x2    14.19/7.92
POL(c9(x1)) = x1    14.19/7.92
POL(false) = 0    14.19/7.92
POL(le(x1, x2)) = 0    14.19/7.92
POL(s(x1)) = [1] + x1    14.19/7.92
POL(true) = [3]   
14.19/7.92
14.19/7.92

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → s(s(0)) 14.19/7.92
p(s(z0)) → z0 14.19/7.92
p(p(s(z0))) → p(z0) 14.19/7.92
le(p(s(z0)), z0) → le(z0, z0) 14.19/7.92
le(0, z0) → true 14.19/7.92
le(s(z0), 0) → false 14.19/7.92
le(s(z0), s(z1)) → le(z0, z1) 14.19/7.92
minus(z0, z1) → if(le(z0, z1), z0, z1) 14.19/7.92
if(true, z0, z1) → 0 14.19/7.92
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c6(LE(z0, z1)) 14.19/7.92
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.92
MINUS(0, z0) → c7 14.19/7.92
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0)) 14.19/7.92
IF(false, s(z0), x1) → c9(MINUS(z0, x1))
S tuples:none
K tuples:

MINUS(0, z0) → c7 14.19/7.92
IF(false, s(z0), x1) → c9(MINUS(z0, x1)) 14.19/7.92
MINUS(s(z0), s(z1)) → c7(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1))) 14.19/7.92
MINUS(s(z0), 0) → c7(IF(false, s(z0), 0)) 14.19/7.92
LE(s(z0), s(z1)) → c6(LE(z0, z1))
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c6, c7, c7, c7, c9

14.19/7.92
14.19/7.92

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

The set S is empty
14.19/7.92
14.19/7.92

(26) BOUNDS(O(1), O(1))

14.19/7.92
14.19/7.92
14.39/8.04 EOF