### (0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

ge, minus, id_inc, div, if

Defined Pair Symbols:

GE, MINUS, ID_INC, DIV, IF

Compound Symbols:

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

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

Removed 8 trailing nodes:

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

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1) → c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1) → c11(ID_INC(div(minus(z0, z1), z1)), DIV(minus(z0, z1), z1), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:

ge, minus, id_inc, div, if

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c8, c11

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

Removed 1 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1) → c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:

ge, minus, id_inc, div, if

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c8, c11

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
DIV(z0, z1) → c8(IF(ge(z1, s(0)), ge(z0, z1), z0, z1), GE(z1, s(0)), GE(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, DIV, IF

Compound Symbols:

c2, c5, c8, c11

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

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

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

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(0, s(z0)) → c8(IF(ge(s(z0), s(0)), false, 0, s(z0)), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0), GE(0, s(0)), GE(z0, 0))
DIV(x0, 0) → c8(IF(false, ge(x0, 0), x0, 0), GE(0, s(0)), GE(x0, 0))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
K tuples:none
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8

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

Removed 1 trailing nodes:

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

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(0, s(z0)) → c8(IF(ge(s(z0), s(0)), false, 0, s(z0)), GE(s(z0), s(0)), GE(0, s(z0)))
DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0), GE(0, s(0)), GE(z0, 0))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
K tuples:none
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8

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

Removed 4 trailing tuple parts

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
K tuples:none
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8

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

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

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
We considered the (Usable) Rules:

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

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(s(z0), s(z1)) → c8(IF(ge(s(z1), s(0)), ge(z0, z1), s(z0), s(z1)), GE(s(z1), s(0)), GE(s(z0), s(z1)))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8

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

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

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

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8

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

Removed 1 trailing tuple parts

### (20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c8

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

Split RHS of tuples not part of any SCC

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c

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

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

DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
We considered the (Usable) Rules:none
And the Tuples:

GE(s(z0), s(z1)) → c2(GE(z0, z1))
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
IF(true, true, z0, z1) → c11(DIV(minus(z0, z1), z1), MINUS(z0, z1))
DIV(x0, s(z0)) → c8(IF(ge(z0, 0), ge(x0, s(z0)), x0, s(z0)), GE(s(z0), s(0)), GE(x0, s(z0)))
DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(z0, 0) → c8(IF(ge(0, s(0)), true, z0, 0))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(s(z1)), s(0)), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z1)), s(0)), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(s(0), s(0)), true, s(z0), s(0)), GE(s(0), s(0)), GE(s(z0), s(0)))
DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(z0), s(0)), GE(s(x0), s(z0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

### (24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c

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

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

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

### (26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c

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

Removed 2 trailing tuple parts

### (28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c

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

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

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

### (30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c

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

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

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

### (32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c

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

Removed 1 trailing nodes:

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

### (34) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

GE, MINUS, IF, DIV

Compound Symbols:

c2, c5, c11, c8, c8, c

### (35) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

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

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

### (36) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(0, s(z0)) → c8(GE(s(z0), s(0)))
DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
DIV(s(0), s(s(z0))) → c(GE(s(s(z0)), s(0)))
DIV(s(0), s(s(z0))) → c(GE(s(0), s(s(z0))))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c11, c8, c8, c, c2

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

Removed 3 trailing nodes:

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

### (38) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c11, c8, c8, c2

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

Removed 6 trailing tuple parts

### (40) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

MINUS, IF, DIV, GE

Compound Symbols:

c5, c11, c8, c2, c8

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

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

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

### (42) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(s(x0), s(x1)) → c8(GE(s(x0), s(x1)))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

IF, DIV, GE, MINUS

Compound Symbols:

c11, c8, c2, c8, c5

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

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

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

### (44) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

IF, GE, DIV, MINUS

Compound Symbols:

c11, c2, c8, c8, c5

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

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

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

### (46) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

IF, DIV, MINUS, GE

Compound Symbols:

c11, c8, c8, c5, c2

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

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

### (48) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

IF, DIV, MINUS, GE

Compound Symbols:

c11, c8, c8, c5, c2

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

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

### (50) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

IF, DIV, MINUS, GE

Compound Symbols:

c11, c8, c5, c8, c2

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

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

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

### (52) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11

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

Removed 1 trailing tuple parts

### (54) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

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

### (56) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
Defined Rule Symbols:

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
We considered the (Usable) Rules:

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

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(s(x0)), s(s(x1))), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

### (58) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

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

### (60) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
We considered the (Usable) Rules:

ge(z0, 0) → true
And the Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

### (62) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
We considered the (Usable) Rules:

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

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(x0), s(0)) → c11(DIV(minus(s(x0), s(0)), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

### (64) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

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

### (66) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))
We considered the (Usable) Rules:

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

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(x0, s(z0)) → c8(IF(true, ge(x0, s(z0)), x0, s(z0)), GE(x0, s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(DIV(minus(s(x0), s(x1)), s(s(x1))), MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(z0), s(0)) → c11(DIV(minus(z0, 0), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

### (68) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

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

### (70) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

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

### (72) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

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

### (74) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

Removed 2 trailing tuple parts

### (76) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

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

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

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

DIV(0, s(z0)) → c8
We considered the (Usable) Rules:none
And the Tuples:

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, x0, s(x1)) → c11(DIV(minus(x0, s(x1)), s(x1)), MINUS(x0, s(x1)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
IF(true, true, s(0), s(0)) → c11(DIV(0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
DIV(0, s(z0)) → c8
The order we found is given by the following interpretation:
Polynomial interpretation :

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

### (78) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

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

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

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

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

### (80) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

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

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

Removed 3 trailing nodes:

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

### (82) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

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

### (84) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
We considered the (Usable) Rules:

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

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

### (86) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
We considered the (Usable) Rules:

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

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

### (88) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
We considered the (Usable) Rules:

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

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

### (90) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

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

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

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

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

### (92) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

DIV(s(x0), s(z0)) → c8(IF(ge(z0, 0), ge(x0, z0), s(x0), s(z0)), GE(s(x0), s(z0)))
MINUS(s(s(y0)), s(s(y1))) → c5(MINUS(s(y0), s(y1)))
DIV(s(s(y0)), s(s(y1))) → c8(GE(s(s(y0)), s(s(y1))))
GE(s(s(s(y0))), s(s(s(y1)))) → c2(GE(s(s(y0)), s(s(y1))))
DIV(s(s(z0)), s(s(z1))) → c8(IF(ge(s(z1), 0), ge(z0, z1), s(s(z0)), s(s(z1))), GE(s(s(z0)), s(s(z1))))
DIV(s(z0), s(0)) → c8(IF(ge(0, 0), true, s(z0), s(0)))
IF(true, true, s(z0), s(z1)) → c11(DIV(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF(true, true, s(s(x0)), s(s(x1))) → c11(MINUS(s(s(x0)), s(s(x1))))
IF(true, true, s(s(z0)), s(s(z1))) → c11(DIV(minus(z0, z1), s(s(z1))), MINUS(s(s(z0)), s(s(z1))))
IF(true, true, s(z0), s(0)) → c11(DIV(z0, s(0)))
DIV(s(z0), s(z1)) → c8(IF(true, ge(z0, z1), s(z0), s(z1)), GE(s(z0), s(z1)))
S tuples:none
K tuples:

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

ge, minus

Defined Pair Symbols:

DIV, MINUS, GE, IF

Compound Symbols:

c8, c5, c8, c2, c11, c11

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

The set S is empty