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

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(z0, 0) → c
GE(0, s(z0)) → c1
GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(z0, 0) → c3
MINUS(0, z0) → c4
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
ID_INC(z0) → c6
ID_INC(z0) → c7
QUOT(z0, z1) → c8(DIV(z0, z1, 0))
DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
IF(false, z0, z1, z2, z3) → c10
IF(true, false, z0, z1, z2) → c11
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1), ID_INC(z2))
K tuples:none
Defined Rule Symbols:

ge, minus, id_inc, quot, div, if

Defined Pair Symbols:

GE, MINUS, ID_INC, QUOT, DIV, IF

Compound Symbols:

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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

QUOT(z0, z1) → c8(DIV(z0, z1, 0))
Removed 8 trailing nodes:

IF(true, false, z0, z1, z2) → c11
IF(false, z0, z1, z2, z3) → c10
GE(z0, 0) → c
GE(0, s(z0)) → c1
MINUS(z0, 0) → c3
MINUS(0, z0) → c4
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)
quot(z0, z1) → div(z0, z1, 0)
div(z0, z1, z2) → if(ge(z1, s(0)), ge(z0, z1), z0, z1, z2)
if(false, z0, z1, z2, z3) → div_by_zero
if(true, false, z0, z1, z2) → z2
if(true, true, z0, z1, z2) → div(minus(z0, z1), z1, id_inc(z2))
Tuples:

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1), ID_INC(z2))
K tuples:none
Defined Rule Symbols:

ge, minus, id_inc, quot, div, if

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12

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

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:

ge, minus, id_inc, quot, div, if

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1, z2) → c9(IF(ge(z1, s(0)), ge(z0, z1), z0, z1, z2), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12

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

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

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(IF(ge(s(z0), s(0)), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2), GE(0, s(0)), GE(z0, 0))
DIV(x0, 0, x2) → c9(IF(false, ge(x0, 0), x0, 0, x2), GE(0, s(0)), GE(x0, 0))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
K tuples:none
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c12, c9

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

Removed 1 trailing nodes:

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

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(IF(ge(s(z0), s(0)), false, 0, s(z0), x2), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2), GE(0, s(0)), GE(z0, 0))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
K tuples:none
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c12, c9

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

Removed 4 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
K tuples:none
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c12, c9, c9

(15) CdtRuleRemovalProof (UPPER BOUND (ADD(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), x2) → c9(GE(s(z0), s(0)))
We considered the (Usable) Rules:none
And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1, z2) → c12(DIV(minus(z0, z1), z1, id_inc(z2)), MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(DIV(x1, x2, x3)) = [2]
POL(GE(x1, x2)) = 0
POL(IF(x1, x2, x3, x4, x5)) = [2]
POL(MINUS(x1, x2)) = 0
POL(c12(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(false) = [3]
POL(ge(x1, x2)) = [3] + [2]x1 + [4]x2
POL(id_inc(x1)) = [3] + [3]x1
POL(minus(x1, x2)) = [5] + [2]x1 + [3]x2
POL(s(x1)) = 0
POL(true) = [3]

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c12, c9, c9

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

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

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

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c9, c12

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

Removed 2 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c9, c12, c12

(21) CdtRuleRemovalProof (UPPER BOUND (ADD(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), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
We considered the (Usable) Rules:

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(z0), s(z1), x2) → c9(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(DIV(x1, x2, x3)) = [2]x1
POL(GE(x1, x2)) = 0
POL(IF(x1, x2, x3, x4, x5)) = [2]x3
POL(MINUS(x1, x2)) = 0
POL(c12(x1)) = x1
POL(c12(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(false) = [2]
POL(ge(x1, x2)) = 0
POL(id_inc(x1)) = [3] + [2]x1
POL(minus(x1, x2)) = x1
POL(s(x1)) = [1] + x1
POL(true) = 0

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c9, c12, c12

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

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

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

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c9, c12, c12

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

Removed 1 trailing tuple parts

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

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

Split RHS of tuples not part of any SCC

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

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

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

DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
We considered the (Usable) Rules:

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]
POL(DIV(x1, x2, x3)) = x1
POL(GE(x1, x2)) = 0
POL(IF(x1, x2, x3, x4, x5)) = x3
POL(MINUS(x1, x2)) = 0
POL(c(x1)) = x1
POL(c12(x1)) = x1
POL(c12(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(false) = [3]
POL(ge(x1, x2)) = [5] + [2]x1 + [2]x2
POL(id_inc(x1)) = [1]
POL(minus(x1, x2)) = x1
POL(s(x1)) = x1
POL(true) = [1]

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

(31) CdtRuleRemovalProof (UPPER BOUND (ADD(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), x2) → c9(GE(s(x0), s(x1)))
We considered the (Usable) Rules:

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(x0, s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0), x2), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(z0, 0, x2) → c9(IF(ge(0, s(0)), true, z0, 0, x2))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
IF(true, true, s(z0), s(z1), x2) → c12(DIV(minus(z0, z1), s(z1), id_inc(x2)), MINUS(s(z0), s(z1)))
IF(true, true, z0, 0, x2) → c12(DIV(z0, 0, id_inc(x2)))
IF(true, true, 0, z0, x2) → c12(DIV(0, z0, id_inc(x2)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [2]
POL(DIV(x1, x2, x3)) = x1
POL(GE(x1, x2)) = 0
POL(IF(x1, x2, x3, x4, x5)) = x3
POL(MINUS(x1, x2)) = 0
POL(c(x1)) = x1
POL(c12(x1)) = x1
POL(c12(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(false) = [1]
POL(ge(x1, x2)) = [3]x1
POL(id_inc(x1)) = [4] + [3]x1
POL(minus(x1, x2)) = x1
POL(s(x1)) = [4] + x1
POL(true) = 0

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

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

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

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

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

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

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

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

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

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

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

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

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

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

Removed 2 trailing nodes:

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

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

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

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

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

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

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

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

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

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

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

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

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

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

(49) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

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

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c9, c

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

Removed 2 trailing tuple parts

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c9, c12, c9, c

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

Split RHS of tuples not part of any SCC

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

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

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

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

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

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

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

DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
We considered the (Usable) Rules:none
And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, z0), MINUS(x0, x1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
IF(true, true, s(0), s(z0), x2) → c1(DIV(0, s(z0), id_inc(x2)))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(DIV(x1, x2, x3)) = x2
POL(GE(x1, x2)) = 0
POL(IF(x1, x2, x3, x4, x5)) = x4
POL(MINUS(x1, x2)) = 0
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c12(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(false) = [3]
POL(ge(x1, x2)) = [2]x1
POL(id_inc(x1)) = [2]
POL(minus(x1, x2)) = [1] + [5]x1 + [2]x2
POL(s(x1)) = [1]
POL(true) = [4]

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

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

(59) 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)))

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0), x2) → c9(GE(s(z0), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0)), x2) → c(GE(s(0), s(s(z0))))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
DIV(x0, s(x1), x2) → c9(GE(s(x1), s(0)))
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, DIV, IF, GE

Compound Symbols:

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

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

Removed 5 trailing nodes:

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

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c12, c9, c9, c1, c2

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

Removed 6 trailing tuple parts

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c12, c9, c1, c2, c9

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

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

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

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c12, c9, c1, c2, c9

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

Removed 2 trailing nodes:

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

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus, id_inc

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c12, c9, c1, c2, c9

(69) 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(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
We considered the (Usable) Rules:

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

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, x0, x1, z0) → c12(DIV(minus(x0, x1), x1, s(z0)), MINUS(x0, x1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(DIV(x1, x2, x3)) = x1
POL(GE(x1, x2)) = 0
POL(IF(x1, x2, x3, x4, x5)) = x3
POL(MINUS(x1, x2)) = 0
POL(c1(x1)) = x1
POL(c12(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(c9(x1, x2)) = x1 + x2
POL(false) = [3]
POL(ge(x1, x2)) = [2]x1
POL(id_inc(x1)) = [5]
POL(minus(x1, x2)) = x1
POL(s(x1)) = [2] + x1
POL(true) = 0

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c12, c9, c1, c2, c9

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

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

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

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

MINUS, DIV, IF, GE

Compound Symbols:

c5, c9, c12, c1, c2, c9

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

Removed 2 trailing nodes:

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

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

MINUS, DIV, IF, GE

Compound Symbols:

c5, c9, c12, c1, c2, c9

(75) 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(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
We considered the (Usable) Rules:

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

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

MINUS, DIV, IF, GE

Compound Symbols:

c5, c9, c12, c1, c2, c9

(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(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0
id_inc(z0) → s(z0)
id_inc(z0) → z0
minus(0, z0) → 0
And the Tuples:

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(DIV(x1, x2, x3)) = [1] + [4]x1 + [3]x3
POL(GE(x1, x2)) = 0
POL(IF(x1, x2, x3, x4, x5)) = [4]x3 + [3]x5
POL(MINUS(x1, x2)) = [3]
POL(c1(x1)) = x1
POL(c12(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(c9(x1, x2)) = x1 + x2
POL(false) = [3]
POL(ge(x1, x2)) = x1
POL(id_inc(x1)) = [4] + x1
POL(minus(x1, x2)) = x1
POL(s(x1)) = [4] + x1
POL(true) = 0

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

MINUS, DIV, IF, GE

Compound Symbols:

c5, c9, c12, c1, c2, c9

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

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

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
We considered the (Usable) Rules:

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

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus, id_inc

Defined Pair Symbols:

MINUS, DIV, IF, GE

Compound Symbols:

c5, c9, c12, c1, c2, c9

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

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

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [3]
POL(DIV(x1, x2, x3)) = x2 + x1·x2
POL(GE(x1, x2)) = x2
POL(IF(x1, x2, x3, x4, x5)) = x3·x4
POL(MINUS(x1, x2)) = 0
POL(c1(x1)) = x1
POL(c12(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(c9(x1, x2)) = x1 + x2
POL(false) = 0
POL(ge(x1, x2)) = 0
POL(id_inc(x1)) = 0
POL(minus(x1, x2)) = x1
POL(s(x1)) = [1] + x1
POL(true) = 0

(82) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(s(x0), s(x1), x2) → c9(GE(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), z0), MINUS(s(x0), s(x1)))
IF(true, true, s(x0), s(x1), z0) → c12(DIV(minus(x0, x1), s(x1), s(z0)), MINUS(s(x0), s(x1)))
IF(true, true, s(z0), s(0), x2) → c12(DIV(z0, s(0), id_inc(x2)), MINUS(s(z0), s(0)))
IF(true, true, s(s(z0)), s(s(z1)), x2) → c12(DIV(minus(z0, z1), s(s(z1)), id_inc(x2)), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(0), s(z0), x2) → c1(MINUS(s(0), s(z0)))
GE(s(s(y0)), s(s(y1))) → c2(GE(s(y0), s(y1)))
DIV(s(s(z0)), s(s(z1)), x2) → c9(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0), x2) → c9(IF(ge(s(0), s(0)), true, s(z0), s(0), x2))
DIV(s(x0), s(z0), x2) → c9(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0), x2), GE(s(x0), s(z0)))
DIV(s(z0), s(z1), x2) → c9(IF(true, ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
S tuples:none
K tuples:

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

ge, minus, id_inc

Defined Pair Symbols:

MINUS, DIV, IF, GE

Compound Symbols:

c5, c9, c12, c1, c2, c9

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

The set S is empty