YES(O(1), O(n^2)) 126.08/42.57 YES(O(1), O(n^2)) 126.08/42.58 126.08/42.58 126.08/42.58 126.08/42.58 126.08/42.58 126.08/42.58 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 126.08/42.58 126.08/42.58 126.08/42.58
126.08/42.58
126.08/42.58
126.08/42.58
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
126.08/42.58 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳2 CdtProblem
126.08/42.58 
↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳4 CdtProblem
126.08/42.58 
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳6 CdtProblem
126.08/42.58 
↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳8 CdtProblem
126.08/42.58 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
126.08/42.58 
↳10 CdtProblem
126.08/42.58 
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳12 CdtProblem
126.08/42.58 
↳13 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳14 CdtProblem
126.08/42.58 
↳15 CdtLeafRemovalProof (ComplexityIfPolyImplication)
126.08/42.58 
↳16 CdtProblem
126.08/42.58 
↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
126.08/42.58 
↳18 CdtProblem
126.08/42.58 
↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳20 CdtProblem
126.08/42.58 
↳21 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳22 CdtProblem
126.08/42.58 
↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳24 CdtProblem
126.08/42.58 
↳25 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳26 CdtProblem
126.08/42.58 
↳27 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳28 CdtProblem
126.08/42.58 
↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
126.08/42.58 
↳30 CdtProblem
126.08/42.58 
↳31 CdtInstantiationProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳32 CdtProblem
126.08/42.58 
↳33 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳34 CdtProblem
126.08/42.58 
↳35 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳36 CdtProblem
126.08/42.58 
↳37 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳38 CdtProblem
126.08/42.58 
↳39 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
126.08/42.58 
↳40 CdtProblem
126.08/42.58 
↳41 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳42 CdtProblem
126.08/42.58 
↳43 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳44 CdtProblem
126.08/42.58 
↳45 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
126.08/42.58 
↳46 CdtProblem
126.08/42.58 
↳47 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
126.08/42.58 
↳48 CdtProblem
126.08/42.58 
↳49 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
126.08/42.58 
↳50 CdtProblem
126.08/42.58 
↳51 SIsEmptyProof (BOTH BOUNDS(ID, ID))
126.08/42.58 
↳52 BOUNDS(O(1), O(1))
126.08/42.58
126.08/42.58 126.08/42.58
126.08/42.58
126.08/42.58

(0) Obligation:

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

ge(x, 0) → true 126.08/42.58
ge(0, s(x)) → false 126.08/42.58
ge(s(x), s(y)) → ge(x, y) 126.08/42.58
minus(x, 0) → x 126.08/42.58
minus(s(x), s(y)) → minus(x, y) 126.08/42.58
div(x, y) → ify(ge(y, s(0)), x, y) 126.08/42.58
ify(false, x, y) → divByZeroError 126.08/42.58
ify(true, x, y) → if(ge(x, y), x, y) 126.08/42.58
if(false, x, y) → 0 126.08/42.58
if(true, x, y) → s(div(minus(x, y), y))

Rewrite Strategy: INNERMOST
126.08/42.58
126.08/42.58

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

Converted CpxTRS to CDT
126.08/42.58
126.08/42.58

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.08/42.58
ge(0, s(z0)) → false 126.08/42.58
ge(s(z0), s(z1)) → ge(z0, z1) 126.08/42.58
minus(z0, 0) → z0 126.08/42.58
minus(s(z0), s(z1)) → minus(z0, z1) 126.08/42.58
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.08/42.58
ify(false, z0, z1) → divByZeroError 126.08/42.58
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.08/42.58
if(false, z0, z1) → 0 126.08/42.58
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.08/42.58
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.08/42.58
DIV(z0, z1) → c5(IFY(ge(z1, s(0)), z0, z1), GE(z1, s(0))) 126.08/42.58
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1)) 126.08/42.58
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.08/42.58
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.08/42.58
DIV(z0, z1) → c5(IFY(ge(z1, s(0)), z0, z1), GE(z1, s(0))) 126.08/42.58
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1)) 126.08/42.58
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, DIV, IFY, IF

Compound Symbols:

c2, c4, c5, c7, c9

126.08/42.58
126.08/42.58

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

Use narrowing to replace DIV(z0, z1) → c5(IFY(ge(z1, s(0)), z0, z1), GE(z1, s(0))) by

DIV(x0, 0) → c5(IFY(false, x0, 0), GE(0, s(0))) 126.08/42.58
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0)))
126.08/42.58
126.08/42.58

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.08/42.58
ge(0, s(z0)) → false 126.08/42.58
ge(s(z0), s(z1)) → ge(z0, z1) 126.08/42.58
minus(z0, 0) → z0 126.08/42.58
minus(s(z0), s(z1)) → minus(z0, z1) 126.08/42.58
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.08/42.58
ify(false, z0, z1) → divByZeroError 126.08/42.58
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.08/42.58
if(false, z0, z1) → 0 126.08/42.58
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.08/42.58
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.08/42.58
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1)) 126.08/42.58
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.08/42.58
DIV(x0, 0) → c5(IFY(false, x0, 0), GE(0, s(0))) 126.08/42.58
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0)))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.08/42.58
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.08/42.58
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1)) 126.08/42.58
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.63
DIV(x0, 0) → c5(IFY(false, x0, 0), GE(0, s(0))) 126.72/42.63
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0)))
K tuples:none
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IFY, IF, DIV

Compound Symbols:

c2, c4, c7, c9, c5

126.72/42.63
126.72/42.63

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

Removed 2 trailing tuple parts
126.72/42.63
126.72/42.63

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.63
ge(0, s(z0)) → false 126.72/42.63
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.63
minus(z0, 0) → z0 126.72/42.63
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.63
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.63
ify(false, z0, z1) → divByZeroError 126.72/42.63
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.63
if(false, z0, z1) → 0 126.72/42.63
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.63
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.63
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1)) 126.72/42.63
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.63
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.63
DIV(x0, 0) → c5
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.63
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.63
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1)) 126.72/42.63
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.63
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.63
DIV(x0, 0) → c5
K tuples:none
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IFY, IF, DIV

Compound Symbols:

c2, c4, c7, c9, c5, c5

126.72/42.63
126.72/42.63

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

Removed 1 trailing nodes:

DIV(x0, 0) → c5
126.72/42.63
126.72/42.63

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.63
ge(0, s(z0)) → false 126.72/42.63
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.63
minus(z0, 0) → z0 126.72/42.63
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.63
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.63
ify(false, z0, z1) → divByZeroError 126.72/42.63
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.63
if(false, z0, z1) → 0 126.72/42.63
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.63
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.63
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1)) 126.72/42.63
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.63
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.63
DIV(x0, 0) → c5
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.63
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.63
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1)) 126.72/42.63
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.63
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.63
DIV(x0, 0) → c5
K tuples:none
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IFY, IF, DIV

Compound Symbols:

c2, c4, c7, c9, c5, c5

126.72/42.63
126.72/42.63

(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.

DIV(x0, 0) → c5
We considered the (Usable) Rules:

ge(z0, 0) → true 126.72/42.63
ge(0, s(z0)) → false 126.72/42.63
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.63
minus(z0, 0) → z0 126.72/42.63
minus(s(z0), s(z1)) → minus(z0, z1)
And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.63
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.63
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1)) 126.72/42.63
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.63
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.63
DIV(x0, 0) → c5
The order we found is given by the following interpretation:
Polynomial interpretation : 126.72/42.63

POL(0) = [2]    126.72/42.63
POL(DIV(x1, x2)) = x2    126.72/42.63
POL(GE(x1, x2)) = 0    126.72/42.63
POL(IF(x1, x2, x3)) = x3    126.72/42.63
POL(IFY(x1, x2, x3)) = x3    126.72/42.63
POL(MINUS(x1, x2)) = 0    126.72/42.63
POL(c2(x1)) = x1    126.72/42.63
POL(c4(x1)) = x1    126.72/42.63
POL(c5) = 0    126.72/42.63
POL(c5(x1, x2)) = x1 + x2    126.72/42.63
POL(c7(x1, x2)) = x1 + x2    126.72/42.63
POL(c9(x1, x2)) = x1 + x2    126.72/42.63
POL(false) = [5]    126.72/42.63
POL(ge(x1, x2)) = [5]x2    126.72/42.63
POL(minus(x1, x2)) = [1] + x2    126.72/42.63
POL(s(x1)) = [2] + x1    126.72/42.63
POL(true) = 0   
126.72/42.63
126.72/42.63

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.63
ge(0, s(z0)) → false 126.72/42.63
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.63
minus(z0, 0) → z0 126.72/42.63
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.63
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.63
ify(false, z0, z1) → divByZeroError 126.72/42.63
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.63
if(false, z0, z1) → 0 126.72/42.63
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.63
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.63
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1)) 126.72/42.63
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.63
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.63
DIV(x0, 0) → c5
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.63
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.63
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1)) 126.72/42.63
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.63
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0)))
K tuples:

DIV(x0, 0) → c5
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IFY, IF, DIV

Compound Symbols:

c2, c4, c7, c9, c5, c5

126.72/42.63
126.72/42.63

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

Use narrowing to replace IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1)) by

IFY(true, z0, 0) → c7(IF(true, z0, 0), GE(z0, 0)) 126.72/42.63
IFY(true, 0, s(z0)) → c7(IF(false, 0, s(z0)), GE(0, s(z0))) 126.72/42.63
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
126.72/42.63
126.72/42.63

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.63
ge(0, s(z0)) → false 126.72/42.63
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.63
minus(z0, 0) → z0 126.72/42.63
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.63
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.63
ify(false, z0, z1) → divByZeroError 126.72/42.63
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.63
if(false, z0, z1) → 0 126.72/42.63
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.63
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.63
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.63
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.63
DIV(x0, 0) → c5 126.72/42.63
IFY(true, z0, 0) → c7(IF(true, z0, 0), GE(z0, 0)) 126.72/42.63
IFY(true, 0, s(z0)) → c7(IF(false, 0, s(z0)), GE(0, s(z0))) 126.72/42.63
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.63
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.63
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, z0, 0) → c7(IF(true, z0, 0), GE(z0, 0)) 126.72/42.64
IFY(true, 0, s(z0)) → c7(IF(false, 0, s(z0)), GE(0, s(z0))) 126.72/42.64
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
K tuples:

DIV(x0, 0) → c5
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c5, c7

126.72/42.64
126.72/42.64

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

Removed 3 trailing tuple parts
126.72/42.64
126.72/42.64

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.64
ge(0, s(z0)) → false 126.72/42.64
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.64
minus(z0, 0) → z0 126.72/42.64
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.64
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.64
ify(false, z0, z1) → divByZeroError 126.72/42.64
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.64
if(false, z0, z1) → 0 126.72/42.64
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
DIV(x0, 0) → c5 126.72/42.64
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1))) 126.72/42.64
IFY(true, z0, 0) → c7(IF(true, z0, 0)) 126.72/42.64
IFY(true, 0, s(z0)) → c7
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1))) 126.72/42.64
IFY(true, z0, 0) → c7(IF(true, z0, 0)) 126.72/42.64
IFY(true, 0, s(z0)) → c7
K tuples:

DIV(x0, 0) → c5
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c5, c7, c7, c7

126.72/42.64
126.72/42.64

(15) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

IFY(true, z0, 0) → c7(IF(true, z0, 0))
Removed 2 trailing nodes:

DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7
126.72/42.64
126.72/42.64

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.64
ge(0, s(z0)) → false 126.72/42.64
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.64
minus(z0, 0) → z0 126.72/42.64
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.64
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.64
ify(false, z0, z1) → divByZeroError 126.72/42.64
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.64
if(false, z0, z1) → 0 126.72/42.64
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
DIV(x0, 0) → c5 126.72/42.64
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1))) 126.72/42.64
IFY(true, 0, s(z0)) → c7
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1))) 126.72/42.64
IFY(true, 0, s(z0)) → c7
K tuples:

DIV(x0, 0) → c5
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c5, c7, c7

126.72/42.64
126.72/42.64

(17) 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.

IFY(true, 0, s(z0)) → c7
We considered the (Usable) Rules:

ge(z0, 0) → true 126.72/42.64
ge(0, s(z0)) → false 126.72/42.64
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.64
minus(z0, 0) → z0 126.72/42.64
minus(s(z0), s(z1)) → minus(z0, z1)
And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
DIV(x0, 0) → c5 126.72/42.64
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1))) 126.72/42.64
IFY(true, 0, s(z0)) → c7
The order we found is given by the following interpretation:
Polynomial interpretation : 126.72/42.64

POL(0) = 0    126.72/42.64
POL(DIV(x1, x2)) = [1]    126.72/42.64
POL(GE(x1, x2)) = 0    126.72/42.64
POL(IF(x1, x2, x3)) = [1]    126.72/42.64
POL(IFY(x1, x2, x3)) = [1]    126.72/42.64
POL(MINUS(x1, x2)) = 0    126.72/42.64
POL(c2(x1)) = x1    126.72/42.64
POL(c4(x1)) = x1    126.72/42.64
POL(c5) = 0    126.72/42.64
POL(c5(x1, x2)) = x1 + x2    126.72/42.64
POL(c7) = 0    126.72/42.64
POL(c7(x1, x2)) = x1 + x2    126.72/42.64
POL(c9(x1, x2)) = x1 + x2    126.72/42.64
POL(false) = [3]    126.72/42.64
POL(ge(x1, x2)) = 0    126.72/42.64
POL(minus(x1, x2)) = [1]    126.72/42.64
POL(s(x1)) = 0    126.72/42.64
POL(true) = 0   
126.72/42.64
126.72/42.64

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.64
ge(0, s(z0)) → false 126.72/42.64
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.64
minus(z0, 0) → z0 126.72/42.64
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.64
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.64
ify(false, z0, z1) → divByZeroError 126.72/42.64
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.64
if(false, z0, z1) → 0 126.72/42.64
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
DIV(x0, 0) → c5 126.72/42.64
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1))) 126.72/42.64
IFY(true, 0, s(z0)) → c7
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
K tuples:

DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c5, c7, c7

126.72/42.64
126.72/42.64

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

Use narrowing to replace DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0))) by

DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0)))
126.72/42.64
126.72/42.64

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.64
ge(0, s(z0)) → false 126.72/42.64
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.64
minus(z0, 0) → z0 126.72/42.64
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.64
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.64
ify(false, z0, z1) → divByZeroError 126.72/42.64
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.64
if(false, z0, z1) → 0 126.72/42.64
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, 0) → c5 126.72/42.64
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1))) 126.72/42.64
IFY(true, 0, s(z0)) → c7 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0)))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1))) 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0)))
K tuples:

DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c7, c7, c5

126.72/42.64
126.72/42.64

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

Removed 2 trailing nodes:

DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7
126.72/42.64
126.72/42.64

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.64
ge(0, s(z0)) → false 126.72/42.64
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.64
minus(z0, 0) → z0 126.72/42.64
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.64
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.64
ify(false, z0, z1) → divByZeroError 126.72/42.64
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.64
if(false, z0, z1) → 0 126.72/42.64
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, 0) → c5 126.72/42.64
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1))) 126.72/42.64
IFY(true, 0, s(z0)) → c7 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0)))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1))) 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0)))
K tuples:

DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c7, c7, c5

126.72/42.64
126.72/42.64

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

Use narrowing to replace IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1))) by

IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(0), s(s(z0))) → c7(IF(false, s(0), s(s(z0))), GE(s(0), s(s(z0)))) 126.72/42.64
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.64
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
126.72/42.64
126.72/42.64

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.64
ge(0, s(z0)) → false 126.72/42.64
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.64
minus(z0, 0) → z0 126.72/42.64
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.64
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.64
ify(false, z0, z1) → divByZeroError 126.72/42.64
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.64
if(false, z0, z1) → 0 126.72/42.64
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(0), s(s(z0))) → c7(IF(false, s(0), s(s(z0))), GE(s(0), s(s(z0)))) 126.72/42.64
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.64
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(0), s(s(z0))) → c7(IF(false, s(0), s(s(z0))), GE(s(0), s(s(z0)))) 126.72/42.64
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.64
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
K tuples:

DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c7, c5, c7, c7

126.72/42.64
126.72/42.64

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

Removed 1 trailing tuple parts
126.72/42.64
126.72/42.64

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.64
ge(0, s(z0)) → false 126.72/42.64
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.64
minus(z0, 0) → z0 126.72/42.64
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.64
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.64
ify(false, z0, z1) → divByZeroError 126.72/42.64
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.64
if(false, z0, z1) → 0 126.72/42.64
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.64
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.64
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.64
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.64
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
K tuples:

DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c7, c5, c7, c7

126.72/42.64
126.72/42.64

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

Removed 2 trailing nodes:

DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7
126.72/42.64
126.72/42.64

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.64
ge(0, s(z0)) → false 126.72/42.64
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.64
minus(z0, 0) → z0 126.72/42.64
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.64
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.64
ify(false, z0, z1) → divByZeroError 126.72/42.64
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.64
if(false, z0, z1) → 0 126.72/42.64
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.64
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.64
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.64
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.64
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
K tuples:

DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c7, c5, c7, c7

126.72/42.64
126.72/42.64

(29) 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.

IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.64
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
We considered the (Usable) Rules:

ge(z0, 0) → true 126.72/42.64
ge(0, s(z0)) → false 126.72/42.64
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.64
minus(z0, 0) → z0 126.72/42.64
minus(s(z0), s(z1)) → minus(z0, z1)
And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.64
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.64
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation : 126.72/42.64

POL(0) = 0    126.72/42.64
POL(DIV(x1, x2)) = [1]    126.72/42.64
POL(GE(x1, x2)) = 0    126.72/42.64
POL(IF(x1, x2, x3)) = x1 + [2]x3    126.72/42.64
POL(IFY(x1, x2, x3)) = [1]    126.72/42.64
POL(MINUS(x1, x2)) = 0    126.72/42.64
POL(c2(x1)) = x1    126.72/42.64
POL(c4(x1)) = x1    126.72/42.64
POL(c5) = 0    126.72/42.64
POL(c5(x1, x2)) = x1 + x2    126.72/42.64
POL(c7) = 0    126.72/42.64
POL(c7(x1)) = x1    126.72/42.64
POL(c7(x1, x2)) = x1 + x2    126.72/42.64
POL(c9(x1, x2)) = x1 + x2    126.72/42.64
POL(false) = [1]    126.72/42.64
POL(ge(x1, x2)) = [1]    126.72/42.64
POL(minus(x1, x2)) = [1] + x2    126.72/42.64
POL(s(x1)) = 0    126.72/42.64
POL(true) = [1]   
126.72/42.64
126.72/42.64

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.64
ge(0, s(z0)) → false 126.72/42.64
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.64
minus(z0, 0) → z0 126.72/42.64
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.64
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.64
ify(false, z0, z1) → divByZeroError 126.72/42.64
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.64
if(false, z0, z1) → 0 126.72/42.64
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.64
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.64
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
K tuples:

DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7 126.72/42.64
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.64
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c7, c5, c7, c7

126.72/42.64
126.72/42.64

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

Use instantiation to replace IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1)) by

IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)), MINUS(s(x0), s(0))) 126.72/42.64
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
126.72/42.64
126.72/42.64

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.64
ge(0, s(z0)) → false 126.72/42.64
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.64
minus(z0, 0) → z0 126.72/42.64
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.64
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.64
ify(false, z0, z1) → divByZeroError 126.72/42.64
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.64
if(false, z0, z1) → 0 126.72/42.64
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.64
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.64
DIV(x0, 0) → c5 126.72/42.64
IFY(true, 0, s(z0)) → c7 126.72/42.64
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.64
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.64
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.64
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)), MINUS(s(x0), s(0))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)), MINUS(s(x0), s(0))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
K tuples:

DIV(x0, 0) → c5 126.72/42.65
IFY(true, 0, s(z0)) → c7 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, DIV, IFY, IF

Compound Symbols:

c2, c4, c5, c7, c5, c7, c7, c9

126.72/42.65
126.72/42.65

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

Removed 2 trailing nodes:

DIV(x0, 0) → c5 126.72/42.65
IFY(true, 0, s(z0)) → c7
126.72/42.65
126.72/42.65

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.65
ge(0, s(z0)) → false 126.72/42.65
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.65
minus(z0, 0) → z0 126.72/42.65
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.65
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.65
ify(false, z0, z1) → divByZeroError 126.72/42.65
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.65
if(false, z0, z1) → 0 126.72/42.65
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
IFY(true, 0, s(z0)) → c7 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)), MINUS(s(x0), s(0))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)), MINUS(s(x0), s(0))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
K tuples:

IFY(true, 0, s(z0)) → c7 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IFY, DIV, IF

Compound Symbols:

c2, c4, c7, c5, c7, c7, c9

126.72/42.65
126.72/42.65

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

Use narrowing to replace IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)), MINUS(s(x0), s(0))) by

IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0)))
126.72/42.65
126.72/42.65

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.65
ge(0, s(z0)) → false 126.72/42.65
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.65
minus(z0, 0) → z0 126.72/42.65
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.65
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.65
ify(false, z0, z1) → divByZeroError 126.72/42.65
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.65
if(false, z0, z1) → 0 126.72/42.65
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
IFY(true, 0, s(z0)) → c7 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0)))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0)))
K tuples:

IFY(true, 0, s(z0)) → c7 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IFY, DIV, IF

Compound Symbols:

c2, c4, c7, c5, c7, c7, c9

126.72/42.65
126.72/42.65

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

Removed 1 trailing nodes:

IFY(true, 0, s(z0)) → c7
126.72/42.65
126.72/42.65

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.65
ge(0, s(z0)) → false 126.72/42.65
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.65
minus(z0, 0) → z0 126.72/42.65
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.65
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.65
ify(false, z0, z1) → divByZeroError 126.72/42.65
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.65
if(false, z0, z1) → 0 126.72/42.65
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
IFY(true, 0, s(z0)) → c7 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0)))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0)))
K tuples:

IFY(true, 0, s(z0)) → c7 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IFY, DIV, IF

Compound Symbols:

c2, c4, c7, c5, c7, c7, c9

126.72/42.65
126.72/42.65

(39) 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), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0)))
We considered the (Usable) Rules:

minus(z0, 0) → z0 126.72/42.65
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.65
ge(z0, 0) → true 126.72/42.65
ge(0, s(z0)) → false 126.72/42.65
ge(s(z0), s(z1)) → ge(z0, z1)
And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
IFY(true, 0, s(z0)) → c7 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation : 126.72/42.65

POL(0) = 0    126.72/42.65
POL(DIV(x1, x2)) = [2]x1    126.72/42.65
POL(GE(x1, x2)) = 0    126.72/42.65
POL(IF(x1, x2, x3)) = [2]x2    126.72/42.65
POL(IFY(x1, x2, x3)) = x1 + [2]x2    126.72/42.65
POL(MINUS(x1, x2)) = 0    126.72/42.65
POL(c2(x1)) = x1    126.72/42.65
POL(c4(x1)) = x1    126.72/42.65
POL(c5(x1, x2)) = x1 + x2    126.72/42.65
POL(c7) = 0    126.72/42.65
POL(c7(x1)) = x1    126.72/42.65
POL(c7(x1, x2)) = x1 + x2    126.72/42.65
POL(c9(x1, x2)) = x1 + x2    126.72/42.65
POL(false) = [3]    126.72/42.65
POL(ge(x1, x2)) = 0    126.72/42.65
POL(minus(x1, x2)) = x1    126.72/42.65
POL(s(x1)) = [4] + x1    126.72/42.65
POL(true) = 0   
126.72/42.65
126.72/42.65

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.65
ge(0, s(z0)) → false 126.72/42.65
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.65
minus(z0, 0) → z0 126.72/42.65
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.65
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.65
ify(false, z0, z1) → divByZeroError 126.72/42.65
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.65
if(false, z0, z1) → 0 126.72/42.65
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
IFY(true, 0, s(z0)) → c7 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0)))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
K tuples:

IFY(true, 0, s(z0)) → c7 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0)))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IFY, DIV, IF

Compound Symbols:

c2, c4, c7, c5, c7, c7, c9

126.72/42.65
126.72/42.65

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

Use narrowing to replace IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1)))) by

IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
126.72/42.65
126.72/42.65

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.65
ge(0, s(z0)) → false 126.72/42.65
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.65
minus(z0, 0) → z0 126.72/42.65
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.65
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.65
ify(false, z0, z1) → divByZeroError 126.72/42.65
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.65
if(false, z0, z1) → 0 126.72/42.65
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
IFY(true, 0, s(z0)) → c7 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
K tuples:

IFY(true, 0, s(z0)) → c7 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0)))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IFY, DIV, IF

Compound Symbols:

c2, c4, c7, c5, c7, c7, c9

126.72/42.65
126.72/42.65

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

Removed 1 trailing nodes:

IFY(true, 0, s(z0)) → c7
126.72/42.65
126.72/42.65

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.65
ge(0, s(z0)) → false 126.72/42.65
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.65
minus(z0, 0) → z0 126.72/42.65
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.65
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.65
ify(false, z0, z1) → divByZeroError 126.72/42.65
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.65
if(false, z0, z1) → 0 126.72/42.65
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
IFY(true, 0, s(z0)) → c7 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
K tuples:

IFY(true, 0, s(z0)) → c7 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0)))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IFY, DIV, IF

Compound Symbols:

c2, c4, c7, c5, c7, c7, c9

126.72/42.65
126.72/42.65

(45) 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.

DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.65
minus(z0, 0) → z0 126.72/42.65
ge(z0, 0) → true 126.72/42.65
ge(0, s(z0)) → false 126.72/42.65
ge(s(z0), s(z1)) → ge(z0, z1)
And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
IFY(true, 0, s(z0)) → c7 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
The order we found is given by the following interpretation:
Polynomial interpretation : 126.72/42.65

POL(0) = [4]    126.72/42.65
POL(DIV(x1, x2)) = [5] + [5]x1 + [5]x2    126.72/42.65
POL(GE(x1, x2)) = 0    126.72/42.65
POL(IF(x1, x2, x3)) = [3] + [5]x2 + [5]x3    126.72/42.65
POL(IFY(x1, x2, x3)) = [4] + x1 + [5]x2 + [5]x3    126.72/42.65
POL(MINUS(x1, x2)) = [5]    126.72/42.65
POL(c2(x1)) = x1    126.72/42.65
POL(c4(x1)) = x1    126.72/42.65
POL(c5(x1, x2)) = x1 + x2    126.72/42.65
POL(c7) = 0    126.72/42.65
POL(c7(x1)) = x1    126.72/42.65
POL(c7(x1, x2)) = x1 + x2    126.72/42.65
POL(c9(x1, x2)) = x1 + x2    126.72/42.65
POL(false) = [3]    126.72/42.65
POL(ge(x1, x2)) = 0    126.72/42.65
POL(minus(x1, x2)) = [2] + x1    126.72/42.65
POL(s(x1)) = [4] + x1    126.72/42.65
POL(true) = 0   
126.72/42.65
126.72/42.65

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.65
ge(0, s(z0)) → false 126.72/42.65
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.65
minus(z0, 0) → z0 126.72/42.65
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.65
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.65
ify(false, z0, z1) → divByZeroError 126.72/42.65
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.65
if(false, z0, z1) → 0 126.72/42.65
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
IFY(true, 0, s(z0)) → c7 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
K tuples:

IFY(true, 0, s(z0)) → c7 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0))) 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IFY, DIV, IF

Compound Symbols:

c2, c4, c7, c5, c7, c7, c9

126.72/42.65
126.72/42.65

(47) 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.

MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.65
minus(z0, 0) → z0 126.72/42.65
ge(z0, 0) → true 126.72/42.65
ge(0, s(z0)) → false 126.72/42.65
ge(s(z0), s(z1)) → ge(z0, z1)
And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
IFY(true, 0, s(z0)) → c7 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
The order we found is given by the following interpretation:
Polynomial interpretation : 126.72/42.65

POL(0) = 0    126.72/42.65
POL(DIV(x1, x2)) = [2]x12    126.72/42.65
POL(GE(x1, x2)) = 0    126.72/42.65
POL(IF(x1, x2, x3)) = [2]x22    126.72/42.65
POL(IFY(x1, x2, x3)) = [3]x1 + [3]x12 + [2]x22    126.72/42.65
POL(MINUS(x1, x2)) = [1] + x1    126.72/42.65
POL(c2(x1)) = x1    126.72/42.65
POL(c4(x1)) = x1    126.72/42.65
POL(c5(x1, x2)) = x1 + x2    126.72/42.65
POL(c7) = 0    126.72/42.65
POL(c7(x1)) = x1    126.72/42.65
POL(c7(x1, x2)) = x1 + x2    126.72/42.65
POL(c9(x1, x2)) = x1 + x2    126.72/42.65
POL(false) = [3]    126.72/42.65
POL(ge(x1, x2)) = 0    126.72/42.65
POL(minus(x1, x2)) = x1    126.72/42.65
POL(s(x1)) = [1] + x1    126.72/42.65
POL(true) = 0   
126.72/42.65
126.72/42.65

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.65
ge(0, s(z0)) → false 126.72/42.65
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.65
minus(z0, 0) → z0 126.72/42.65
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.65
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.65
ify(false, z0, z1) → divByZeroError 126.72/42.65
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.65
if(false, z0, z1) → 0 126.72/42.65
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
IFY(true, 0, s(z0)) → c7 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
S tuples:

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

IFY(true, 0, s(z0)) → c7 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0))) 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1)))) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IFY, DIV, IF

Compound Symbols:

c2, c4, c7, c5, c7, c7, c9

126.72/42.65
126.72/42.65

(49) 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.

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

minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.65
minus(z0, 0) → z0 126.72/42.65
ge(z0, 0) → true 126.72/42.65
ge(0, s(z0)) → false 126.72/42.65
ge(s(z0), s(z1)) → ge(z0, z1)
And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
IFY(true, 0, s(z0)) → c7 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
The order we found is given by the following interpretation:
Polynomial interpretation : 126.72/42.65

POL(0) = 0    126.72/42.65
POL(DIV(x1, x2)) = [2] + x1 + x2 + [2]x1·x2    126.72/42.65
POL(GE(x1, x2)) = x2    126.72/42.65
POL(IF(x1, x2, x3)) = x2 + [2]x2·x3    126.72/42.65
POL(IFY(x1, x2, x3)) = [3]x1 + x2 + x3 + [2]x2·x3 + [3]x12    126.72/42.65
POL(MINUS(x1, x2)) = [3]x2    126.72/42.65
POL(c2(x1)) = x1    126.72/42.65
POL(c4(x1)) = x1    126.72/42.65
POL(c5(x1, x2)) = x1 + x2    126.72/42.65
POL(c7) = 0    126.72/42.65
POL(c7(x1)) = x1    126.72/42.65
POL(c7(x1, x2)) = x1 + x2    126.72/42.65
POL(c9(x1, x2)) = x1 + x2    126.72/42.65
POL(false) = [3]    126.72/42.65
POL(ge(x1, x2)) = 0    126.72/42.65
POL(minus(x1, x2)) = x1    126.72/42.65
POL(s(x1)) = [2] + x1    126.72/42.65
POL(true) = 0   
126.72/42.65
126.72/42.65

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(z0, 0) → true 126.72/42.65
ge(0, s(z0)) → false 126.72/42.65
ge(s(z0), s(z1)) → ge(z0, z1) 126.72/42.65
minus(z0, 0) → z0 126.72/42.65
minus(s(z0), s(z1)) → minus(z0, z1) 126.72/42.65
div(z0, z1) → ify(ge(z1, s(0)), z0, z1) 126.72/42.65
ify(false, z0, z1) → divByZeroError 126.72/42.65
ify(true, z0, z1) → if(ge(z0, z1), z0, z1) 126.72/42.65
if(false, z0, z1) → 0 126.72/42.65
if(true, z0, z1) → s(div(minus(z0, z1), z1))
Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1)) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
IFY(true, 0, s(z0)) → c7 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
S tuples:none
K tuples:

IFY(true, 0, s(z0)) → c7 126.72/42.65
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1))) 126.72/42.65
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0)))) 126.72/42.65
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)), MINUS(s(z0), s(0))) 126.72/42.65
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0))) 126.72/42.65
IFY(true, s(s(z0)), s(s(z1))) → c7(IF(ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) 126.72/42.65
IF(true, s(s(x0)), s(s(x1))) → c9(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1)))) 126.72/42.65
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 126.72/42.65
GE(s(z0), s(z1)) → c2(GE(z0, z1))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IFY, DIV, IF

Compound Symbols:

c2, c4, c7, c5, c7, c7, c9

126.72/42.65
126.72/42.65

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

The set S is empty
126.72/42.65
126.72/42.65

(52) BOUNDS(O(1), O(1))

126.72/42.65
126.72/42.65
126.99/42.70 EOF