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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 11 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtUsableRulesProof (⇔, 0 ms)
↳6 CdtProblem
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CdtProblem
↳9 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CdtProblem
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CdtProblem
↳13 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳14 CdtProblem
↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CdtProblem
↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳18 CdtProblem
↳19 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳20 CdtProblem
↳21 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 51 ms)
↳22 CdtProblem
↳23 CdtInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳24 CdtProblem
↳25 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳26 CdtProblem
↳27 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 11 ms)
↳28 CdtProblem
↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳30 CdtProblem
↳31 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 17 ms)
↳32 CdtProblem
↳33 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)
↳34 CdtProblem
↳35 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 163 ms)
↳36 CdtProblem
↳37 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 214 ms)
↳38 CdtProblem
↳39 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳40 BOUNDS(1, 1)

(0) Obligation:

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


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 Cpx (relative) TRS 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(z0, 0) → c
GE(0, s(z0)) → c1
GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(z0, 0) → c3
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(false, z0, z1) → c6
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1))
IF(false, z0, z1) → c8
IF(true, z0, z1) → c9(DIV(minus(z0, z1), z1), MINUS(z0, z1))
S tuples:

GE(z0, 0) → c
GE(0, s(z0)) → c1
GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(z0, 0) → c3
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(false, z0, z1) → c6
IFY(true, z0, z1) → c7(IF(ge(z0, z1), z0, z1), GE(z0, z1))
IF(false, z0, z1) → c8
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:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

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

Removed 5 trailing nodes:

IF(false, z0, z1) → c8
MINUS(z0, 0) → c3
IFY(false, z0, z1) → c6
GE(0, s(z0)) → c1
GE(z0, 0) → c

(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))
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

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(0, s(z0)) → false
ge(s(z0), s(z1)) → ge(z0, z1)
ge(z0, 0) → true
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, 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

Defined Pair Symbols:

GE, MINUS, DIV, IFY, IF

Compound Symbols:

c2, c4, c5, c7, c9

(7) 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)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(0, s(z0)) → false
ge(s(z0), s(z1)) → ge(z0, z1)
ge(z0, 0) → true
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, 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

Defined Pair Symbols:

GE, MINUS, IFY, IF, DIV

Compound Symbols:

c2, c4, c7, c9, c5

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

Removed 1 trailing nodes:

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(0, s(z0)) → false
ge(s(z0), s(z1)) → ge(z0, z1)
ge(z0, 0) → true
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, 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

Defined Pair Symbols:

GE, MINUS, IFY, IF, DIV

Compound Symbols:

c2, c4, c7, c9, c5

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

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c7

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

IFY(true, z0, 0) → c7(IF(true, z0, 0), GE(z0, 0))
Removed 1 trailing nodes:

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

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(0, s(z0)) → false
ge(s(z0), s(z1)) → ge(z0, z1)
ge(z0, 0) → true
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, 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

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c7

(15) 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)))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(0, s(z0)) → false
ge(s(z0), s(z1)) → ge(z0, z1)
ge(z0, 0) → true
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, 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

Defined Pair Symbols:

GE, MINUS, IF, IFY, DIV

Compound Symbols:

c2, c4, c9, c7, c5

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

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(0, s(z0)) → false
ge(s(z0), s(z1)) → ge(z0, z1)
ge(z0, 0) → true
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, 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(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(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
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(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(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
K tuples:none
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c7, c7

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

Removed 1 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(0, s(z0)) → false
ge(s(z0), s(z1)) → ge(z0, z1)
ge(z0, 0) → true
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, 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(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(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
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))
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(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(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
K tuples:none
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c7, c7

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(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))) → c7(GE(s(0), s(s(z0))))
We considered the (Usable) Rules:none
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(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(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
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 :

POL(0) = 0   
POL(DIV(x1, x2)) = x2   
POL(GE(x1, x2)) = 0   
POL(IF(x1, x2, x3)) = x3   
POL(IFY(x1, x2, x3)) = x3   
POL(MINUS(x1, x2)) = 0   
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) = 0   
POL(ge(x1, x2)) = 0   
POL(minus(x1, x2)) = 0   
POL(s(x1)) = [1]   
POL(true) = 0   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

ge(0, s(z0)) → false
ge(s(z0), s(z1)) → ge(z0, z1)
ge(z0, 0) → true
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, 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(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(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
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))
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(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(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
K tuples:

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

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV, IFY

Compound Symbols:

c2, c4, c9, c5, c7, c7

(23) 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)))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

ge, minus

Defined Pair Symbols:

GE, MINUS, DIV, IFY, IF

Compound Symbols:

c2, c4, c5, c7, c7, c9

(25) 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))))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

ge, minus

Defined Pair Symbols:

GE, MINUS, DIV, IFY, IF

Compound Symbols:

c2, c4, c5, c7, c7, c9

(27) CdtRuleRemovalProof (UPPER BOUND(ADD(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
And the Tuples:

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

POL(0) = 0   
POL(DIV(x1, x2)) = x1 + x2   
POL(GE(x1, x2)) = 0   
POL(IF(x1, x2, x3)) = x2 + x3   
POL(IFY(x1, x2, x3)) = x2 + x3   
POL(MINUS(x1, x2)) = 0   
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) = 0   
POL(ge(x1, x2)) = 0   
POL(minus(x1, x2)) = x1   
POL(s(x1)) = [1] + x1   
POL(true) = 0   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

Defined Pair Symbols:

GE, MINUS, DIV, IFY, IF

Compound Symbols:

c2, c4, c5, c7, c7, c9

(29) 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)))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

Defined Pair Symbols:

GE, MINUS, DIV, IFY, IF

Compound Symbols:

c2, c4, c5, c7, c7, c9

(31) CdtRuleRemovalProof (UPPER BOUND(ADD(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(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0
And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), 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(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
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)), MINUS(s(z0), 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)) = 0   
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) = 0   
POL(ge(x1, x2)) = 0   
POL(minus(x1, x2)) = x1   
POL(s(x1)) = [1] + x1   
POL(true) = 0   

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

GE, MINUS, DIV, IFY, IF

Compound Symbols:

c2, c4, c5, c7, c7, c9

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

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

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

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

GE, MINUS, DIV, IFY, IF

Compound Symbols:

c2, c4, c5, c7, c7, c9

(35) CdtRuleRemovalProof (UPPER BOUND(ADD(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)
minus(z0, 0) → z0
And the Tuples:

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

POL(0) = [1]   
POL(DIV(x1, x2)) = [2] + [2]x12   
POL(GE(x1, x2)) = 0   
POL(IF(x1, x2, x3)) = [2]x22   
POL(IFY(x1, x2, x3)) = [1] + x1 + x12 + [2]x22   
POL(MINUS(x1, x2)) = [1] + [2]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) = [2]   
POL(ge(x1, x2)) = 0   
POL(minus(x1, x2)) = x1   
POL(s(x1)) = [2] + x1   
POL(true) = 0   

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
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)), MINUS(s(z0), s(0)))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), 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(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, DIV, IFY, IF

Compound Symbols:

c2, c4, c5, c7, c7, c9

(37) CdtRuleRemovalProof (UPPER BOUND(ADD(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:

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
DIV(x0, s(z0)) → c5(IFY(true, x0, s(z0)), 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(z0), s(0)) → c7(IF(true, s(z0), s(0)), GE(s(z0), s(0)))
IFY(true, s(x0), s(x1)) → c7(GE(s(x0), s(x1)))
IFY(true, s(0), s(s(z0))) → c7(GE(s(0), s(s(z0))))
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)), MINUS(s(z0), 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]x2 + [2]x1·x2 + [2]x12   
POL(GE(x1, x2)) = x2   
POL(IF(x1, x2, x3)) = [2]x2·x3 + [2]x12 + [2]x1·x2 + [2]x22   
POL(IFY(x1, x2, x3)) = [2]x1 + [2]x2·x3 + x1·x3 + [2]x1·x2 + [2]x22   
POL(MINUS(x1, x2)) = 0   
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)) = x1   
POL(s(x1)) = [1] + x1   
POL(true) = [1]   

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

ge, minus

Defined Pair Symbols:

GE, MINUS, DIV, IFY, IF

Compound Symbols:

c2, c4, c5, c7, c7, c9

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

The set S is empty

(40) BOUNDS(1, 1)