Runtime Complexity (innermost) proof of /tmp/tmpEhDlLN/thiemann26.xml

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), 10 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)

↳4 CdtProblem

↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CdtProblem

↳7 CdtUsableRulesProof (⇔, 0 ms)

↳8 CdtProblem

↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳10 CdtProblem

↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳12 CdtProblem

↳13 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳14 CdtProblem

↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 62 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)), 30 ms)

↳22 CdtProblem

↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳24 CdtProblem

↳25 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳26 CdtProblem

↳27 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳28 CdtProblem

↳29 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 15 ms)

↳30 CdtProblem

↳31 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 78 ms)

↳32 CdtProblem

↳33 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳34 CdtProblem

↳35 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳36 CdtProblem

↳37 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)

↳38 CdtProblem

↳39 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳40 CdtProblem

↳41 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳42 CdtProblem

↳43 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳44 CdtProblem

↳45 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳46 CdtProblem

↳47 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 17 ms)

↳48 CdtProblem

↳49 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)

↳50 CdtProblem

↳51 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳52 CdtProblem

↳53 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳54 CdtProblem

↳55 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)

↳56 CdtProblem

↳57 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 41 ms)

↳58 CdtProblem

↳59 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳60 CdtProblem

↳61 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)

↳62 CdtProblem

↳63 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳64 CdtProblem

↳65 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳66 CdtProblem

↳67 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳68 CdtProblem

↳69 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳70 CdtProblem

↳71 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 24 ms)

↳72 CdtProblem

↳73 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳74 CdtProblem

↳75 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳76 CdtProblem

↳77 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 10 ms)

↳78 CdtProblem

↳79 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 21 ms)

↳80 CdtProblem

↳81 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 898 ms)

↳82 CdtProblem

↳83 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 1784 ms)

↳84 CdtProblem

↳85 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)

↳86 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(y)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0) → x
minus(0, y) → 0
minus(s(x), s(y)) → minus(x, y)
id_inc(x) → x
id_inc(x) → s(x)
quot(x, y) → div(x, y, 0)
div(x, y, z) → if(ge(y, s(0)), ge(x, y), x, y, z)
if(false, b, x, y, z) → div_by_zero
if(true, false, x, y, z) → z
if(true, true, x, y, z) → div(minus(x, y), y, id_inc(z))

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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)
quot(z0, z1) → div(z0, z1, 0)
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
if(false, z0, z1, z2, z3) → div_by_zero
if(true, false, z0, z1, z2) → z2
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))

Tuples:

GE(z0, 0) → c
GE(0, s(z0)) → c1
GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(z0, 0) → c3
MINUS(0, z0) → c4
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
ID_INC(z0) → c6
ID_INC(z0) → c7
QUOT(z0, z1) → c8(DIV(z0, z1, 0))
DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
IF(false, z0, z1, z2, z3) → c10
IF(true, false, z0, z1, z2) → c11
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1), ID_INC(z2))

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(0, z0) → c4
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
ID_INC(z0) → c6
ID_INC(z0) → c7
QUOT(z0, z1) → c8(DIV(z0, z1, 0))
DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
IF(false, z0, z1, z2, z3) → c10
IF(true, false, z0, z1, z2) → c11
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1), ID_INC(z2))

K tuples:none
Defined Rule Symbols:

ge, minus, id_inc, quot, div, if

Defined Pair Symbols:

GE, MINUS, ID_INC, QUOT, DIV, IF

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

QUOT(z0, z1) → c8(DIV(z0, z1, 0))

Removed 8 trailing nodes:

MINUS(0, z0) → c4
GE(0, s(z0)) → c1
ID_INC(z0) → c7
ID_INC(z0) → c6
IF(true, false, z0, z1, z2) → c11
GE(z0, 0) → c
IF(false, z0, z1, z2, z3) → c10
MINUS(z0, 0) → c3

(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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)
quot(z0, z1) → div(z0, z1, 0)
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
if(false, z0, z1, z2, z3) → div_by_zero
if(true, false, z0, z1, z2) → z2
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1), ID_INC(z2))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1), ID_INC(z2))

K tuples:none
Defined Rule Symbols:

ge, minus, id_inc, quot, div, if

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12

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

Removed 1 trailing tuple parts

(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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)
quot(z0, z1) → div(z0, z1, 0)
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
if(false, z0, z1, z2, z3) → div_by_zero
if(true, false, z0, z1, z2) → z2
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))

K tuples:none
Defined Rule Symbols:

ge, minus, id_inc, quot, div, if

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

quot(z0, z1) → div(z0, z1, 0)
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
if(false, z0, z1, z2, z3) → div_by_zero
if(true, false, z0, z1, z2) → z2
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))

(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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))

K tuples:none
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12

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

Use narrowing to replace DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1)) by

DIV(0, s(z0), x2) → c9(IF(ge(s(z0), s(0)), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2), GE(0, s(0)), GE(z0, 0))
DIV(x0, 0, x2) → c9(IF(false, ge(x0, 0), x0, 0, x2), GE(0, s(0)), GE(x0, 0))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))

(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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(IF(ge(s(z0), s(0)), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2), GE(0, s(0)), GE(z0, 0))
DIV(x0, 0, x2) → c9(IF(false, ge(x0, 0), x0, 0, x2), GE(0, s(0)), GE(x0, 0))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(IF(ge(s(z0), s(0)), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2), GE(0, s(0)), GE(z0, 0))
DIV(x0, 0, x2) → c9(IF(false, ge(x0, 0), x0, 0, x2), GE(0, s(0)), GE(x0, 0))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))

K tuples:none
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c12, c9

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

Removed 1 trailing nodes:

DIV(x0, 0, x2) → c9(IF(false, ge(x0, 0), x0, 0, x2), GE(0, s(0)), GE(x0, 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(IF(ge(s(z0), s(0)), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2), GE(0, s(0)), GE(z0, 0))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(IF(ge(s(z0), s(0)), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2), GE(0, s(0)), GE(z0, 0))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))

K tuples:none
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c12, c9

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

Removed 4 trailing tuple parts

(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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))

K tuples:none
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c12, c9, c9

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))

We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0
minus(0, z0) → 0

And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]
POL(DIV(x₁, x₂, x₃)) = x₁
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄, x₅)) = x₃
POL(MINUS(x₁, x₂)) = 0
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(false) = 0
POL(ge(x₁, x₂)) = 0
POL(id_inc(x₁)) = 0
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = x₁
POL(true) = 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c12, c9, c9

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

Use narrowing to replace IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1)) by

IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)), MINUS(z0, 0))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)), MINUS(0, z0))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))

(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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)), MINUS(z0, 0))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)), MINUS(0, z0))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)), MINUS(z0, 0))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)), MINUS(0, z0))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c9, c12

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

Removed 2 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c9, c12, c12

(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.

IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))

We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0
minus(0, z0) → 0

And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(DIV(x₁, x₂, x₃)) = x₁
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄, x₅)) = x₃
POL(MINUS(x₁, x₂)) = 0
POL(c12(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(false) = 0
POL(ge(x₁, x₂)) = 0
POL(id_inc(x₁)) = 0
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = [1] + x₁
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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c9, c12, c12

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

Use narrowing to replace DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1))) by

DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(0), s(s(z0)), x2) → c9(IF(ge(s(s(z0)), s(0)), false, s(0), s(s(z0)), x2), GE(s(s(z0)), s(0)), GE(s(0), s(s(z0))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

(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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(0), s(s(z0)), x2) → c9(IF(ge(s(s(z0)), s(0)), false, s(0), s(s(z0)), x2), GE(s(s(z0)), s(0)), GE(s(0), s(s(z0))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(0), s(s(z0)), x2) → c9(IF(ge(s(s(z0)), s(0)), false, s(0), s(s(z0)), x2), GE(s(s(z0)), s(0)), GE(s(0), s(s(z0))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c9, c12, c12

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

Removed 1 trailing tuple parts

(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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c9(GE(s(s(z0)), s(0)), GE(s(0), s(s(z0))))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c9(GE(s(s(z0)), s(0)), GE(s(0), s(s(z0))))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c9, c12, c12, c9

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

Split RHS of tuples not part of any SCC

(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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c9, c12, c12, c

(29) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))

We considered the (Usable) Rules:

id_inc(z0) → s(z0)
id_inc(z0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0
minus(0, z0) → 0

And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]
POL(DIV(x₁, x₂, x₃)) = x₁ + x₃
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄, x₅)) = x₃ + x₅
POL(MINUS(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(false) = 0
POL(ge(x₁, x₂)) = 0
POL(id_inc(x₁)) = x₁
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = x₁
POL(true) = 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c9, c12, c12, c

(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.

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0
minus(0, z0) → 0

And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(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, x₃)) = x₁
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄, x₅)) = x₃
POL(MINUS(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(false) = 0
POL(ge(x₁, x₂)) = 0
POL(id_inc(x₁)) = 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c9, c12, c12, c

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

Use narrowing to replace DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0))) by

DIV(0, s(z0), x2) → c9(IF(ge(z0, 0), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(ge(z1, 0), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))

(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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(0, s(z0), x2) → c9(IF(ge(z0, 0), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(0, s(z0), x2) → c9(IF(ge(z0, 0), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c12, c9, c

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

Removed 2 trailing tuple parts

(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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c12, c9, c

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

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

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))

(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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c12, c9, c

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

Use narrowing to replace DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2)) by

DIV(x0, 0, x1) → c9(IF(false, true, x0, 0, x1))

(40) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(x0, 0, x1) → c9(IF(false, true, x0, 0, x1))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(x0, 0, x1) → c9(IF(false, true, x0, 0, x1))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c12, c9, c

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

Removed 2 trailing nodes:

DIV(x0, 0, x1) → c9(IF(false, true, x0, 0, x1))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))

(42) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c12, c9, c

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

Use narrowing to replace IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1))) by

IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))

(44) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c12, c9, c

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

Use narrowing to replace IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2))) by

IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))

(46) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c9, c, c12

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

Use narrowing to replace DIV(x0, s(z0), x2) → c9(IF(true, ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0))) by

DIV(0, s(z0), x2) → c9(IF(true, false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))

(48) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))
DIV(0, s(z0), x2) → c9(IF(true, false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))
DIV(0, s(z0), x2) → c9(IF(true, false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c9, c, c12

(49) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

IF(true, true, 0, x0, z0) → c12(DIV(0, x0, z0))
IF(true, true, 0, x0, z0) → c12(DIV(0, x0, s(z0)))

(50) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
DIV(0, s(z0), x2) → c9(IF(true, false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(0, s(z0), x2) → c9(IF(true, false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c9, c

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

Removed 2 trailing tuple parts

(52) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(0), s(z0), x2) → c12(DIV(0, s(z0), id_inc(x2)), MINUS(s(0), s(z0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c9, c

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

Split RHS of tuples not part of any SCC

(54) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c9, c, c1

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

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

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))

(56) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c9, c, c1

(57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))

We considered the (Usable) Rules:none
And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(DIV(x₁, x₂, x₃)) = x₂
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄, x₅)) = x₄
POL(MINUS(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(false) = 0
POL(ge(x₁, x₂)) = 0
POL(id_inc(x₁)) = [2] + [2]x₁
POL(minus(x₁, x₂)) = 0
POL(s(x₁)) = [2]
POL(true) = 0

(58) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c9, c, c1

(59) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1)))) by DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))

(60) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c9, c, c1

(61) 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)))

(62) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))

S tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))

K tuples:

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, DIV, IF, GE

Compound Symbols:

c5, c9, c12, c9, c, c1, c2

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

Removed 5 trailing nodes:

IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))

(64) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))

S tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))

K tuples:

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c12, c9, c9, c1, c2

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

Removed 6 trailing tuple parts

(66) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

S tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

K tuples:

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c12, c9, c1, c2, c9

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

Use narrowing to replace IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1)) by

IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, x2), MINUS(z0, 0))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, x2), MINUS(0, z0))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), x2), MINUS(s(z0), s(z1)))

(68) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, x2), MINUS(z0, 0))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, x2), MINUS(0, z0))

S tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, x2), MINUS(z0, 0))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, x2), MINUS(0, z0))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), x2), MINUS(s(z0), s(z1)))

K tuples:

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c12, c9, c1, c2, c9

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

Removed 2 trailing nodes:

IF(true, true, 0, z0, x2) → c12(DIV(0, z0, x2), MINUS(0, z0))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, x2), MINUS(z0, 0))

(70) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

S tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), x2), MINUS(s(z0), s(z1)))

K tuples:

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c12, c9, c1, c2, c9

(71) 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, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))

We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0
minus(0, z0) → 0

And the Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(DIV(x₁, x₂, x₃)) = x₁
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄, x₅)) = x₃
POL(MINUS(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(c9(x₁, x₂)) = x₁ + x₂
POL(false) = 0
POL(ge(x₁, x₂)) = 0
POL(id_inc(x₁)) = 0
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = [1] + x₁
POL(true) = 0

(72) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

S tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

K tuples:

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c12, c9, c1, c2, c9

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

Use narrowing to replace IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1)) by

IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, s(x2)), MINUS(z0, 0))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, s(x2)), MINUS(0, z0))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), s(x2)), MINUS(s(z0), s(z1)))

(74) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, s(x2)), MINUS(z0, 0))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, s(x2)), MINUS(0, z0))

S tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, s(x2)), MINUS(z0, 0))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, s(x2)), MINUS(0, z0))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), s(x2)), MINUS(s(z0), s(z1)))

K tuples:

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, DIV, IF, GE

Compound Symbols:

c5, c9, c12, c1, c2, c9

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

Removed 2 trailing nodes:

IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, s(x2)), MINUS(z0, 0))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, s(x2)), MINUS(0, z0))

(76) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

S tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), s(x2)), MINUS(s(z0), s(z1)))

K tuples:

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, DIV, IF, GE

Compound Symbols:

c5, c9, c12, c1, c2, c9

(77) 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, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))

We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0
minus(0, z0) → 0

And the Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(DIV(x₁, x₂, x₃)) = [1] + x₁
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄, x₅)) = [1] + x₃
POL(MINUS(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(c9(x₁, x₂)) = x₁ + x₂
POL(false) = 0
POL(ge(x₁, x₂)) = 0
POL(id_inc(x₁)) = 0
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = [1] + x₁
POL(true) = 0

(78) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

S tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

K tuples:

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, DIV, IF, GE

Compound Symbols:

c5, c9, c12, c1, c2, c9

(79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0
minus(0, z0) → 0

And the Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(DIV(x₁, x₂, x₃)) = [1] + x₁
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄, x₅)) = x₃
POL(MINUS(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(c9(x₁, x₂)) = x₁ + x₂
POL(false) = 0
POL(ge(x₁, x₂)) = 0
POL(id_inc(x₁)) = 0
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = [1] + x₁
POL(true) = 0

(80) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

S tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))

K tuples:

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, DIV, IF, GE

Compound Symbols:

c5, c9, c12, c1, c2, c9

(81) 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)) → c5(MINUS(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
minus(0, z0) → 0

And the Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [2]
POL(DIV(x₁, x₂, x₃)) = [1] + [2]x₁ + x₁²
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄, x₅)) = [2]x₂·x₃ + x₃²
POL(MINUS(x₁, x₂)) = [2]x₁
POL(c1(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(c9(x₁, x₂)) = x₁ + x₂
POL(false) = [1]
POL(ge(x₁, x₂)) = [1]
POL(id_inc(x₁)) = 0
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = [1] + x₁
POL(true) = [1]

(82) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

S tuples:

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

K tuples:

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, DIV, IF, GE

Compound Symbols:

c5, c9, c12, c1, c2, c9

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

We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0
minus(0, z0) → 0

And the Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(DIV(x₁, x₂, x₃)) = [2]x₁ + [2]x₁²
POL(GE(x₁, x₂)) = [2]x₁
POL(IF(x₁, x₂, x₃, x₄, x₅)) = [2]x₃²
POL(MINUS(x₁, x₂)) = [1]
POL(c1(x₁)) = x₁
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(c9(x₁, x₂)) = x₁ + x₂
POL(false) = [1]
POL(ge(x₁, x₂)) = 0
POL(id_inc(x₁)) = [2]
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = [2] + x₁
POL(true) = 0

(84) 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(0, z0) → 0
minus(s(z0), s(z1)) → minus(z0, z1)
id_inc(z0) → z0
id_inc(z0) → s(z0)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))

S tuples:none
K tuples:

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
DIV(s(s(z0)), s(s(z1)), z2) → c9(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1)), z2), GE(s(s(z0)), s(s(z1))))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))

Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, DIV, IF, GE

Compound Symbols:

c5, c9, c12, c1, c2, c9

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

The set S is empty