The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

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

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
DIV(z0, z1) → c5(IFY(ge(z1, s(0)), z0, z1), GE(z1, s(0)))
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1))
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

(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)))
DIV(x0, s(z0)) → c5(IFY(ge(z0, 0), x0, s(z0)), GE(s(z0), s(0)))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1))
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, 0) → c5(IFY(false, x0, 0), GE(0, s(0)))
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))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1))
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, 0) → c5(IFY(false, x0, 0), GE(0, s(0)))
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

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 6 dangling nodes:

DIV(x0, 0) → c5(IFY(false, x0, 0), GE(0, s(0)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1))
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
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))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1))
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
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

(7) 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))
IFY(true, 0, s(z0)) → c7(IF(false, 0, s(z0)), GE(0, s(z0)))
IFY(true, s(z0), s(z1)) → c7(IF(ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c7

(9) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 7 dangling nodes:

IFY(true, 0, s(z0)) → c7(IF(false, 0, s(z0)), GE(0, s(z0)))
IFY(true, z0, 0) → c7(IF(true, z0, 0), GE(z0, 0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c7

(11) 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)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, IFY, DIV

Compound Symbols:

c2, c4, c9, c7, c5

(13) 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)))
IFY(true, s(0), s(s(z0))) → c7(IF(false, s(0), s(s(z0))), GE(s(0), s(s(z0))))
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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
IFY(true, s(0), s(s(z0))) → c7(IF(false, s(0), s(s(z0))), GE(s(0), s(s(z0))))
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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
K tuples:none
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c7, c7

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

Split RHS of tuples not part of any SCC

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
IFY(true, s(0), s(s(z0))) → c(IF(false, s(0), s(s(z0))))
IFY(true, s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
K tuples:none
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
IFY(true, s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
IFY(true, s(0), s(s(z0))) → c
K tuples:none
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

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

(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.

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

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

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

POL(0) = 0   
POL(DIV(x1, x2)) = [1] + x2   
POL(GE(x1, x2)) = 0   
POL(IF(x1, x2, x3)) = x1 + [4]x3   
POL(IFY(x1, x2, x3)) = x1   
POL(MINUS(x1, x2)) = 0   
POL(c) = 0   
POL(c(x1)) = x1   
POL(c2(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1, x2)) = x1 + x2   
POL(c7(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(false) = [1]   
POL(ge(x1, x2)) = [1]   
POL(minus(x1, x2)) = [4] + [3]x1 + [3]x2   
POL(s(x1)) = 0   
POL(true) = [1]   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
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:

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

ge, minus, div, ify, if

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

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

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

Use forward instantiation to replace GE(s(z0), s(z1)) → c2(GE(z0, z1)) by

GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
IFY(true, s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
IFY(true, s(0), s(s(z0))) → c
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
S tuples:

MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
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))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
K tuples:

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

ge, minus, div, ify, if

Defined Pair Symbols:

MINUS, IF, DIV, IFY, GE

Compound Symbols:

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

(23) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 9 dangling nodes:

IFY(true, s(0), s(s(z0))) → c
IFY(true, s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
S tuples:

MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), GE(s(z0), s(0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
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))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
K tuples:

IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

MINUS, IF, DIV, IFY, GE

Compound Symbols:

c4, c9, c5, c7, c7, c2

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

Removed 2 trailing tuple parts

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
S tuples:

MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
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))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
K tuples:

IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

MINUS, IF, IFY, GE, DIV

Compound Symbols:

c4, c9, c7, c7, c2, c5

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

Use forward instantiation to replace MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) by

MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
S tuples:

IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
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))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
K tuples:

IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

IF, IFY, GE, DIV, MINUS

Compound Symbols:

c9, c7, c7, c2, c5, c4

(29) 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(s(x0)), s(s(x1))) → c9(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)), MINUS(s(x0), s(0)))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
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))))
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)), MINUS(s(x0), s(0)))
S tuples:

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))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
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))))
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)), MINUS(s(x0), s(0)))
K tuples:

IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

IFY, GE, DIV, MINUS, IF

Compound Symbols:

c7, c7, c2, c5, c4, c9

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

Removed 1 trailing tuple parts

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
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))))
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)))
S tuples:

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))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
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))))
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)))
K tuples:

IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

IFY, GE, DIV, MINUS, IF

Compound Symbols:

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

(33) 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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
S tuples:

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))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
K tuples:

IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

IFY, GE, DIV, MINUS, IF

Compound Symbols:

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

(35) 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(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
We considered the (Usable) Rules:

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

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(DIV(x1, x2)) = [1]   
POL(GE(x1, x2)) = [3]x1 + [3]x2   
POL(IF(x1, x2, x3)) = x1 + [2]x2 + [4]x3   
POL(IFY(x1, x2, x3)) = x1   
POL(MINUS(x1, x2)) = [2]x2   
POL(c2(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c7(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c9(x1)) = x1   
POL(c9(x1, x2)) = x1 + x2   
POL(false) = [1]   
POL(ge(x1, x2)) = [1]   
POL(minus(x1, x2)) = [4]   
POL(s(x1)) = 0   
POL(true) = [1]   

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
S tuples:

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))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)))
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, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

IFY, GE, DIV, MINUS, IF

Compound Symbols:

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

(37) 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(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)
minus(z0, 0) → z0
ge(z0, 0) → true
ge(0, s(z0)) → false
ge(s(z0), s(z1)) → ge(z0, z1)
And the Tuples:

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
S tuples:

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))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
IF(true, s(x0), s(0)) → c9(DIV(minus(s(x0), s(0)), s(0)))
K tuples:

IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
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:

IFY, GE, DIV, MINUS, IF

Compound Symbols:

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

(39) 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))) by

IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)))
S tuples:

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))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)))
K tuples:

IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
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:

IFY, GE, DIV, MINUS, IF

Compound Symbols:

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

(41) 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)))
We considered the (Usable) Rules:

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

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)))
S tuples:

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))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
K tuples:

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

ge, minus, div, ify, if

Defined Pair Symbols:

IFY, GE, DIV, MINUS, IF

Compound Symbols:

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

(43) CdtKnowledgeProof (EQUIVALENT transformation)

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

DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)))
S tuples:

GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
K tuples:

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

ge, minus, div, ify, if

Defined Pair Symbols:

IFY, GE, DIV, MINUS, IF

Compound Symbols:

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

(45) 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(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
We considered the (Usable) Rules:

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

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)))
S tuples:

GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
K tuples:

IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
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))))
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
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))))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

IFY, GE, DIV, MINUS, IF

Compound Symbols:

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

(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.

GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
We considered the (Usable) Rules:

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

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
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))))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)))
S tuples:none
K tuples:

IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
IF(true, s(s(x0)), s(s(x1))) → c9(MINUS(s(s(x0)), s(s(x1))))
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))))
IF(true, s(z0), s(0)) → c9(DIV(minus(z0, 0), s(0)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)))
IFY(true, s(z0), s(0)) → c7(IF(true, s(z0), s(0)))
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))))
MINUS(s(s(y0)), s(s(y1))) → c4(MINUS(s(y0), s(y1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
Defined Rule Symbols:

ge, minus, div, ify, if

Defined Pair Symbols:

IFY, GE, DIV, MINUS, IF

Compound Symbols:

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

(49) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(50) BOUNDS(O(1), O(1))