Runtime Complexity (innermost) proof of /tmp/tmpIWxwPk/thiemann28.xml

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

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 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(O(n^1))), 43 ms)

↳16 CdtProblem

↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 3 ms)

↳18 CdtProblem

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

↳20 CdtProblem

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

↳22 CdtProblem

↳23 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

↳24 CdtProblem

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

↳26 CdtProblem

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

↳28 CdtProblem

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

↳30 CdtProblem

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

↳32 CdtProblem

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

↳34 CdtProblem

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

↳36 CdtProblem

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

↳38 CdtProblem

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

↳40 CdtProblem

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

↳42 CdtProblem

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

↳44 CdtProblem

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

↳46 CdtProblem

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

↳48 CdtProblem

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

↳50 CdtProblem

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

↳52 CdtProblem

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

↳54 CdtProblem

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

↳56 CdtProblem

↳57 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 182 ms)

↳58 CdtProblem

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

↳60 CdtProblem

↳61 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 1 ms)

↳62 CdtProblem

↳63 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 540 ms)

↳64 CdtProblem

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

↳66 CdtProblem

↳67 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 4791 ms)

↳68 CdtProblem

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

↳70 CdtProblem

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

↳72 CdtProblem

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

↳74 CdtProblem

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

↳76 CdtProblem

↳77 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 40 ms)

↳78 CdtProblem

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

↳80 CdtProblem

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

↳82 CdtProblem

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

↳84 CdtProblem

↳85 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 145 ms)

↳86 CdtProblem

↳87 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 66 ms)

↳88 CdtProblem

↳89 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))), 522 ms)

↳90 CdtProblem

↳91 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))), 620 ms)

↳92 CdtProblem

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

↳94 BOUNDS(O(1), O(1))

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

ge, minus, id_inc, div, if

Defined Pair Symbols:

GE, MINUS, ID_INC, DIV, IF

Compound Symbols:

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

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

Removed 8 trailing nodes:

IF(true, false, z0, z1) → c10
GE(z0, 0) → c
GE(0, s(z0)) → c1
MINUS(z0, 0) → c3
MINUS(0, z0) → c4
IF(false, z0, z1, z2) → c9
ID_INC(z0) → c6
ID_INC(z0) → c7

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

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1) → c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1) → c11(ID_INC(div(minus(z0, z1), z1)), DIV(minus(z0, z1), z1), 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) → c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1) → c11(ID_INC(div(minus(z0, z1), z1)), DIV(minus(z0, z1), z1), MINUS(z0, z1))

K tuples:none
Defined Rule Symbols:

ge, minus, id_inc, div, if

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c8, c11

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

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1) → c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), 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) → c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))

K tuples:none
Defined Rule Symbols:

ge, minus, id_inc, div, if

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c8, c11

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(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)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1) → c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), 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) → c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))

K tuples:none
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c8, c11

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

Use narrowing to replace DIV(z0, z1) → c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)), GE(z0, z1)) by

DIV(0, s(z0)) → c8(IF(ge(s(z0), s(0)), false, 0, s(z0)), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0), GE(0, s(0)), GE(z0, 0))
DIV(x0, 0) → c8(IF(false, ge(x0, 0), x0, 0), GE(0, s(0)), GE(x0, 0))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), 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)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(0, s(z0)) → c8(IF(ge(s(z0), s(0)), false, 0, s(z0)), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0), GE(0, s(0)), GE(z0, 0))
DIV(x0, 0) → c8(IF(false, ge(x0, 0), x0, 0), GE(0, s(0)), GE(x0, 0))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), 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) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(0, s(z0)) → c8(IF(ge(s(z0), s(0)), false, 0, s(z0)), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0), GE(0, s(0)), GE(z0, 0))
DIV(x0, 0) → c8(IF(false, ge(x0, 0), x0, 0), GE(0, s(0)), GE(x0, 0))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))

K tuples:none
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8

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

Removed 1 trailing nodes:

DIV(x0, 0) → c8(IF(false, ge(x0, 0), x0, 0), 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)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(0, s(z0)) → c8(IF(ge(s(z0), s(0)), false, 0, s(z0)), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0), GE(0, s(0)), GE(z0, 0))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), 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) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(0, s(z0)) → c8(IF(ge(s(z0), s(0)), false, 0, s(z0)), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0), GE(0, s(0)), GE(z0, 0))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))

K tuples:none
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8

(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)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))

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) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))

K tuples:none
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8

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

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

DIV(0, s(z0)) → c8(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) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))

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

POL(0) = [4]
POL(DIV(x₁, x₂)) = [4]x₁
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄)) = [4]x₃
POL(MINUS(x₁, x₂)) = 0
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(false) = [3]
POL(ge(x₁, x₂)) = [4]x₁
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)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))

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) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))

K tuples:

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8

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

Use narrowing to replace DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1))) by

DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(0), s(s(z0))) → c8(IF(ge(s(s(z0)), s(0)), false, s(0), s(s(z0))), GE(s(s(z0)), s(0)), GE(s(0), s(s(z0))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(0), s(s(z0))) → c8(IF(ge(s(s(z0)), s(0)), false, s(0), s(s(z0))), GE(s(s(z0)), s(0)), GE(s(0), s(s(z0))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(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))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(0), s(s(z0))) → c8(IF(ge(s(s(z0)), s(0)), false, s(0), s(s(z0))), GE(s(s(z0)), s(0)), GE(s(0), s(s(z0))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))

K tuples:

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8

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

Removed 1 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c8(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))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c8(GE(s(s(z0)), s(0)), GE(s(0), s(s(z0))))

K tuples:

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c8

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

Split RHS of tuples not part of any SCC

(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)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))

K tuples:

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c

(23) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → 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)) = [4]
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄)) = [4]
POL(MINUS(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(false) = [3]
POL(ge(x₁, x₂)) = [2] + x₂
POL(minus(x₁, x₂)) = [3] + [3]x₁ + [3]x₂
POL(s(x₁)) = 0
POL(true) = 0

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))

S tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))

K tuples:

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c

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

Use narrowing to replace DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0))) by

DIV(0, s(z0)) → c8(IF(ge(z0, 0), false, 0, s(z0)), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1)) → c8(IF(ge(z1, 0), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))

(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)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
DIV(0, s(z0)) → c8(IF(ge(z0, 0), false, 0, s(z0)), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), 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) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(0, s(z0)) → c8(IF(ge(z0, 0), false, 0, s(z0)), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))

K tuples:

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c

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

Removed 2 trailing tuple parts

(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)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), 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) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))

K tuples:

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c

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

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

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), 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) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))

K tuples:

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c

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

Use narrowing to replace DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0)) by

DIV(x0, 0) → c8(IF(false, true, x0, 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)

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(x0, 0) → c8(IF(false, true, x0, 0))

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) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(x0, 0) → c8(IF(false, true, x0, 0))

K tuples:

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c

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

Removed 1 trailing nodes:

DIV(x0, 0) → c8(IF(false, true, x0, 0))

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), 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) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))

K tuples:

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c

(35) 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)))

(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)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
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, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))

K tuples:

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c11, c8, c8, c, c2

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

Removed 3 trailing nodes:

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))

(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)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
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, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))

K tuples:

DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c11, c8, c8, c2

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

Removed 6 trailing tuple parts

(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)

Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))

S tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))

K tuples:

DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c11, c8, c2, c8

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

Use forward instantiation to replace MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) by

MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))

(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)

Tuples:

IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))

S tuples:

IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))

K tuples:

DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

IF, DIV, GE, MINUS

Compound Symbols:

c11, c8, c2, c8, c5

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

Use forward instantiation to replace DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1))) by

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))

(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)

Tuples:

IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))

S tuples:

IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

IF, GE, DIV, MINUS

Compound Symbols:

c11, c2, c8, c8, c5

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

Use forward instantiation to replace GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1))) by

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

(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)

Tuples:

IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))

S tuples:

IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

IF, DIV, MINUS, GE

Compound Symbols:

c11, c8, c8, c5, c2

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

Used rewriting to replace DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1)))) by DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))

(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)

Tuples:

IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))

S tuples:

IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

IF, DIV, MINUS, GE

Compound Symbols:

c11, c8, c8, c5, c2

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

Used rewriting to replace DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0))) by DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))

(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)

Tuples:

IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))

S tuples:

IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

IF, DIV, MINUS, GE

Compound Symbols:

c11, c8, c5, c8, c2

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

Use instantiation to replace IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1)) by

IF(true, true, s(x0), s(x1)) → c11(DIV(minus(s(x0), s(x1)), s(x1)), MINUS(s(x0), s(x1)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)), MINUS(s(x0), s(0)))

(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(x0), s(x1)) → c11(DIV(minus(s(x0), s(x1)), s(x1)), MINUS(s(x0), s(x1)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)), MINUS(s(x0), s(0)))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(x0), s(x1)) → c11(DIV(minus(s(x0), s(x1)), s(x1)), MINUS(s(x0), s(x1)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)), MINUS(s(x0), s(0)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11

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

Removed 1 trailing tuple parts

(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(x0), s(x1)) → c11(DIV(minus(s(x0), s(x1)), s(x1)), MINUS(s(x0), s(x1)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(x0), s(x1)) → c11(DIV(minus(s(x0), s(x1)), s(x1)), MINUS(s(x0), s(x1)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

Use narrowing to replace IF(true, true, s(x0), s(x1)) → c11(DIV(minus(s(x0), s(x1)), s(x1)), MINUS(s(x0), s(x1))) by

IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), 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:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

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

POL(0) = 0
POL(DIV(x₁, x₂)) = [4]x₁
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄)) = [4]x₃
POL(MINUS(x₁, x₂)) = 0
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(false) = [3]
POL(ge(x₁, x₂)) = [5]x₁
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = [4] + x₁
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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

Use narrowing to replace IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1)))) by

IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))

(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

(61) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))

We considered the (Usable) Rules:

ge(z0, 0) → true

And the Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))

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

POL(0) = [3]
POL(DIV(x₁, x₂)) = [1] + [5]x₂
POL(GE(x₁, x₂)) = [2]x₂
POL(IF(x₁, x₂, x₃, x₄)) = x₁
POL(MINUS(x₁, x₂)) = [2]x₂
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(false) = [5]
POL(ge(x₁, x₂)) = [1]
POL(minus(x₁, x₂)) = [2] + [2]x₂
POL(s(x₁)) = 0
POL(true) = [1]

(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

(63) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))

We considered the (Usable) Rules:

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

And the Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))

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

POL(0) = 0
POL(DIV(x₁, x₂)) = x₁
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄)) = x₃
POL(MINUS(x₁, x₂)) = 0
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(false) = [2]
POL(ge(x₁, x₂)) = 0
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = [2] + x₁
POL(true) = 0

(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

Use narrowing to replace IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0))) by

IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))

(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

(67) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), 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:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))

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

POL(0) = 0
POL(DIV(x₁, x₂)) = [2] + x₁
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄)) = [2] + x₃
POL(MINUS(x₁, x₂)) = 0
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(false) = [3]
POL(ge(x₁, x₂)) = 0
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = [2] + x₁
POL(true) = 0

(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

Use narrowing to replace IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1)))) by

IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))

(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

Use narrowing to replace IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0))) by

IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
IF(true, true, s(0), s(0)) → c11(DIV(0, s(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
IF(true, true, s(0), s(0)) → c11(DIV(0, s(0)))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

Use narrowing to replace DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0))) by

DIV(0, s(z0)) → c8(IF(true, false, 0, s(z0)), GE(0, s(z0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
IF(true, true, s(0), s(0)) → c11(DIV(0, s(0)))
DIV(0, s(z0)) → c8(IF(true, false, 0, s(z0)), GE(0, s(z0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
DIV(0, s(z0)) → c8(IF(true, false, 0, s(z0)), GE(0, s(z0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

Removed 2 trailing tuple parts

(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
IF(true, true, s(0), s(0)) → c11(DIV(0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
DIV(0, s(z0)) → c8

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
DIV(0, s(z0)) → c8

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11, c8

(77) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

DIV(0, s(z0)) → c8

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

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
IF(true, true, s(0), s(0)) → c11(DIV(0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
DIV(0, s(z0)) → c8

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

POL(0) = 0
POL(DIV(x₁, x₂)) = [1] + x₂
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄)) = [1]
POL(MINUS(x₁, x₂)) = 0
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c8) = 0
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(false) = [3]
POL(ge(x₁, x₂)) = [3]x₁
POL(minus(x₁, x₂)) = [5] + x₁ + [2]x₂
POL(s(x₁)) = 0
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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
IF(true, true, s(0), s(0)) → c11(DIV(0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
DIV(0, s(z0)) → c8

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))
DIV(0, s(z0)) → c8

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11, c8

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

Use narrowing to replace IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1))) by

IF(true, true, 0, s(x1)) → c11(DIV(0, s(x1)), MINUS(0, s(x1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
IF(true, true, s(0), s(0)) → c11(DIV(0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
DIV(0, s(z0)) → c8
IF(true, true, 0, s(x1)) → c11(DIV(0, s(x1)), MINUS(0, s(x1)))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
IF(true, true, 0, s(x1)) → c11(DIV(0, s(x1)), MINUS(0, s(x1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))
DIV(0, s(z0)) → c8

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11, c8

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

Removed 3 trailing nodes:

IF(true, true, s(0), s(0)) → c11(DIV(0, s(0)))
DIV(0, s(z0)) → c8
IF(true, true, 0, s(x1)) → c11(DIV(0, s(x1)), MINUS(0, s(x1)))

(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

(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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

(85) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, 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:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

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

POL(0) = 0
POL(DIV(x₁, x₂)) = [4] + [4]x₁
POL(GE(x₁, x₂)) = [4]
POL(IF(x₁, x₂, x₃, x₄)) = [4]x₃
POL(MINUS(x₁, x₂)) = [3]
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(false) = [2]
POL(ge(x₁, x₂)) = [4]x₁
POL(minus(x₁, x₂)) = [1] + x₁
POL(s(x₁)) = [4] + x₁
POL(true) = 0

(86) 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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

S tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

(87) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(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:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

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

POL(0) = 0
POL(DIV(x₁, x₂)) = [2] + [3]x₁ + x₂
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄)) = [3]x₃ + x₄
POL(MINUS(x₁, x₂)) = [3]
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(false) = [3]
POL(ge(x₁, x₂)) = [3]x₁
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = [4] + x₁
POL(true) = 0

(88) 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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

S tuples:

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

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

(89) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(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:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

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

POL(0) = 0
POL(DIV(x₁, x₂)) = [2]x₁²
POL(GE(x₁, x₂)) = 0
POL(IF(x₁, x₂, x₃, x₄)) = [2]x₃²
POL(MINUS(x₁, x₂)) = [2]x₁
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(false) = 0
POL(ge(x₁, x₂)) = 0
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = [1] + x₁
POL(true) = 0

(90) 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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

S tuples:

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

K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

(91) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

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

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

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

POL(0) = 0
POL(DIV(x₁, x₂)) = [2]x₁ + [2]x₁²
POL(GE(x₁, x₂)) = [2]x₁
POL(IF(x₁, x₂, x₃, x₄)) = [2]x₃²
POL(MINUS(x₁, x₂)) = [1]
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(false) = 0
POL(ge(x₁, x₂)) = 0
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = [1] + x₁
POL(true) = 0

(92) 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)

Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))

S tuples:none
K tuples:

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))

Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

The set S is empty