### (0) Obligation:

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

cond1(true, x, y) → cond2(gr(x, y), x, y)
cond2(true, x, y) → cond3(gr(x, 0), x, y)
cond2(false, x, y) → cond4(gr(y, 0), x, y)
cond3(true, x, y) → cond3(gr(x, 0), p(x), y)
cond3(false, x, y) → cond1(and(gr(x, 0), gr(y, 0)), x, y)
cond4(true, x, y) → cond4(gr(y, 0), x, p(y))
cond4(false, x, y) → cond1(and(gr(x, 0), gr(y, 0)), x, y)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
and(true, true) → true
and(false, x) → false
and(x, false) → false
p(0) → 0
p(s(x)) → x

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond1(true, z0, z1) → cond2(gr(z0, z1), z0, z1)
cond2(true, z0, z1) → cond3(gr(z0, 0), z0, z1)
cond2(false, z0, z1) → cond4(gr(z1, 0), z0, z1)
cond3(true, z0, z1) → cond3(gr(z0, 0), p(z0), z1)
cond3(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)
cond4(true, z0, z1) → cond4(gr(z1, 0), z0, p(z1))
cond4(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
p(0) → 0
p(s(z0)) → z0
Tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1), GR(z0, 0))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1), GR(z1, 0))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1), GR(z0, 0), P(z0))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)), GR(z1, 0), P(z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
GR(0, z0) → c7
GR(s(z0), 0) → c8
GR(s(z0), s(z1)) → c9(GR(z0, z1))
AND(true, true) → c10
AND(false, z0) → c11
AND(z0, false) → c12
P(0) → c13
P(s(z0)) → c14
S tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1), GR(z0, 0))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1), GR(z1, 0))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1), GR(z0, 0), P(z0))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)), GR(z1, 0), P(z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
GR(0, z0) → c7
GR(s(z0), 0) → c8
GR(s(z0), s(z1)) → c9(GR(z0, z1))
AND(true, true) → c10
AND(false, z0) → c11
AND(z0, false) → c12
P(0) → c13
P(s(z0)) → c14
K tuples:none
Defined Rule Symbols:

cond1, cond2, cond3, cond4, gr, and, p

Defined Pair Symbols:

COND1, COND2, COND3, COND4, GR, AND, P

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14

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

Removed 7 trailing nodes:

AND(false, z0) → c11
AND(true, true) → c10
P(s(z0)) → c14
AND(z0, false) → c12
P(0) → c13
GR(0, z0) → c7
GR(s(z0), 0) → c8

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond1(true, z0, z1) → cond2(gr(z0, z1), z0, z1)
cond2(true, z0, z1) → cond3(gr(z0, 0), z0, z1)
cond2(false, z0, z1) → cond4(gr(z1, 0), z0, z1)
cond3(true, z0, z1) → cond3(gr(z0, 0), p(z0), z1)
cond3(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)
cond4(true, z0, z1) → cond4(gr(z1, 0), z0, p(z1))
cond4(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
p(0) → 0
p(s(z0)) → z0
Tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1), GR(z0, 0))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1), GR(z1, 0))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1), GR(z0, 0), P(z0))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)), GR(z1, 0), P(z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
GR(s(z0), s(z1)) → c9(GR(z0, z1))
S tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1), GR(z0, 0))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1), GR(z1, 0))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1), GR(z0, 0), P(z0))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)), GR(z1, 0), P(z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0), GR(z1, 0))
GR(s(z0), s(z1)) → c9(GR(z0, z1))
K tuples:none
Defined Rule Symbols:

cond1, cond2, cond3, cond4, gr, and, p

Defined Pair Symbols:

COND1, COND2, COND3, COND4, GR

Compound Symbols:

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

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

Removed 12 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond1(true, z0, z1) → cond2(gr(z0, z1), z0, z1)
cond2(true, z0, z1) → cond3(gr(z0, 0), z0, z1)
cond2(false, z0, z1) → cond4(gr(z1, 0), z0, z1)
cond3(true, z0, z1) → cond3(gr(z0, 0), p(z0), z1)
cond3(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)
cond4(true, z0, z1) → cond4(gr(z1, 0), z0, p(z1))
cond4(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
p(0) → 0
p(s(z0)) → z0
Tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
S tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
K tuples:none
Defined Rule Symbols:

cond1, cond2, cond3, cond4, gr, and, p

Defined Pair Symbols:

COND1, GR, COND2, COND3, COND4

Compound Symbols:

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

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

cond1(true, z0, z1) → cond2(gr(z0, z1), z0, z1)
cond2(true, z0, z1) → cond3(gr(z0, 0), z0, z1)
cond2(false, z0, z1) → cond4(gr(z1, 0), z0, z1)
cond3(true, z0, z1) → cond3(gr(z0, 0), p(z0), z1)
cond3(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)
cond4(true, z0, z1) → cond4(gr(z1, 0), z0, p(z1))
cond4(false, z0, z1) → cond1(and(gr(z0, 0), gr(z1, 0)), z0, z1)

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
S tuples:

COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1))
GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
K tuples:none
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

COND1, GR, COND2, COND3, COND4

Compound Symbols:

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

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

Use narrowing to replace COND1(true, z0, z1) → c(COND2(gr(z0, z1), z0, z1), GR(z0, z1)) by

COND1(true, 0, z0) → c(COND2(false, 0, z0), GR(0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0), GR(s(z0), 0))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0), GR(0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0), GR(s(z0), 0))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0), GR(0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0), GR(s(z0), 0))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND2, COND3, COND4, COND1

Compound Symbols:

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

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

Removed 2 trailing tuple parts

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
K tuples:none
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND2, COND3, COND4, COND1

Compound Symbols:

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

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

Use narrowing to replace COND2(true, z0, z1) → c1(COND3(gr(z0, 0), z0, z1)) by

COND2(true, 0, x1) → c1(COND3(false, 0, x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, 0, x1) → c1(COND3(false, 0, x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, 0, x1) → c1(COND3(false, 0, x1))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
K tuples:none
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND2, COND3, COND4, COND1

Compound Symbols:

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

### (15) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

COND2(true, 0, x1) → c1(COND3(false, 0, x1))

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
K tuples:none
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND2, COND3, COND4, COND1

Compound Symbols:

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

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

Use narrowing to replace COND2(false, z0, z1) → c2(COND4(gr(z1, 0), z0, z1)) by

COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
K tuples:none
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND3, COND4, COND1, COND2

Compound Symbols:

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

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

Use narrowing to replace COND3(true, z0, z1) → c3(COND3(gr(z0, 0), p(z0), z1)) by

COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))

### (20) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
K tuples:none
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND3, COND4, COND1, COND2

Compound Symbols:

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

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

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

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
We considered the (Usable) Rules:

p(s(z0)) → z0
p(0) → 0
And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = [4]x2
POL(COND2(x1, x2, x3)) = [4]x2
POL(COND3(x1, x2, x3)) = [4]x2
POL(COND4(x1, x2, x3)) = [4]x2
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = 0

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND3, COND4, COND1, COND2

Compound Symbols:

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

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

Use narrowing to replace COND3(false, z0, z1) → c4(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1)) by

COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))

### (24) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND4, COND1, COND2, COND3

Compound Symbols:

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

### (25) 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.

COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
We considered the (Usable) Rules:

gr(s(z0), 0) → true
and(z0, false) → false
and(false, z0) → false
and(true, true) → true
gr(0, z0) → false
And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = x1
POL(COND2(x1, x2, x3)) = [2]
POL(COND3(x1, x2, x3)) = [2]
POL(COND4(x1, x2, x3)) = [2]
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = x2
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = [2]
POL(p(x1)) = 0
POL(s(x1)) = 0
POL(true) = [2]

### (26) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND4, COND1, COND2, COND3

Compound Symbols:

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

### (27) 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.

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
We considered the (Usable) Rules:

gr(s(z0), 0) → true
and(z0, false) → false
and(false, z0) → false
and(true, true) → true
gr(0, z0) → false
And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [2]
POL(COND1(x1, x2, x3)) = x1
POL(COND2(x1, x2, x3)) = [2]
POL(COND3(x1, x2, x3)) = [2]
POL(COND4(x1, x2, x3)) = [2]
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = x1
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = x2
POL(p(x1)) = 0
POL(s(x1)) = 0
POL(true) = [2]

### (28) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1)))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND4, COND1, COND2, COND3

Compound Symbols:

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

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

Use narrowing to replace COND4(true, z0, z1) → c5(COND4(gr(z1, 0), z0, p(z1))) by

COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))

### (30) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND4, COND1, COND2, COND3

Compound Symbols:

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

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

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

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
We considered the (Usable) Rules:

p(s(z0)) → z0
p(0) → 0
And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = [4]x3
POL(COND2(x1, x2, x3)) = [4]x3
POL(COND3(x1, x2, x3)) = [4]x3
POL(COND4(x1, x2, x3)) = [4]x3
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = 0

### (32) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND4, COND1, COND2, COND3

Compound Symbols:

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

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

Use narrowing to replace COND4(false, z0, z1) → c6(COND1(and(gr(z0, 0), gr(z1, 0)), z0, z1)) by

COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))

### (34) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

### (35) 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.

COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
We considered the (Usable) Rules:

gr(s(z0), 0) → true
and(z0, false) → false
and(false, z0) → false
and(true, true) → true
gr(0, z0) → false
And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = x1
POL(COND2(x1, x2, x3)) = [1]
POL(COND3(x1, x2, x3)) = [1]
POL(COND4(x1, x2, x3)) = [1]
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = x2
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = [1]
POL(p(x1)) = 0
POL(s(x1)) = 0
POL(true) = [1]

### (36) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

### (37) 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.

COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
We considered the (Usable) Rules:

gr(s(z0), 0) → true
and(z0, false) → false
and(false, z0) → false
and(true, true) → true
gr(0, z0) → false
And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = x1
POL(COND2(x1, x2, x3)) = [1]
POL(COND3(x1, x2, x3)) = [1]
POL(COND4(x1, x2, x3)) = [1]
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = x1
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = [1]
POL(p(x1)) = 0
POL(s(x1)) = 0
POL(true) = [1]

### (38) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

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

Use narrowing to replace COND1(true, s(z0), s(z1)) → c(COND2(gr(z0, z1), s(z0), s(z1)), GR(s(z0), s(z1))) by

COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

### (40) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

### (41) 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.

COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
We considered the (Usable) Rules:none
And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [5]
POL(COND1(x1, x2, x3)) = [1]
POL(COND2(x1, x2, x3)) = [1]
POL(COND3(x1, x2, x3)) = [1]
POL(COND4(x1, x2, x3)) = [1]
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = [4]
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1)) = x1
POL(false) = [3]
POL(gr(x1, x2)) = [2] + [3]x1 + [4]x2
POL(p(x1)) = [4] + [4]x1
POL(s(x1)) = [2] + x1
POL(true) = [3]

### (42) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

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

Use narrowing to replace COND3(true, 0, x1) → c3(COND3(gr(0, 0), 0, x1)) by

COND3(true, 0, x0) → c3(COND3(false, 0, x0))

### (44) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

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

Use narrowing to replace COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1)) by

COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))

### (46) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, 0, x1) → c3(COND3(false, p(0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

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

Use narrowing to replace COND3(true, 0, x1) → c3(COND3(false, p(0), x1)) by

COND3(true, 0, x0) → c3(COND3(false, 0, x0))

### (48) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

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

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

COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

### (50) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

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

Use narrowing to replace COND3(true, s(z0), x1) → c3(COND3(true, p(s(z0)), x1)) by

COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))

### (52) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

### (53) 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.

COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
We considered the (Usable) Rules:none
And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [3]
POL(COND1(x1, x2, x3)) = x2
POL(COND2(x1, x2, x3)) = x2
POL(COND3(x1, x2, x3)) = [3]x1 + x2
POL(COND4(x1, x2, x3)) = x2
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = [2] + x1 + [2]x2
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = [3] + [4]x1
POL(p(x1)) = [5]
POL(s(x1)) = [4] + x1
POL(true) = 0

### (54) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

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

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

COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))

### (56) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

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

Use narrowing to replace COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0)) by

COND3(false, x0, 0) → c4(COND1(false, x0, 0))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))

### (58) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, x0, 0) → c4(COND1(false, x0, 0))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND3(true, s(z0), x1) → c3(COND3(gr(s(z0), 0), z0, x1))
COND3(false, x0, 0) → c4(COND1(and(gr(x0, 0), false), x0, 0))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

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

Removed 1 trailing nodes:

COND3(false, x0, 0) → c4(COND1(false, x0, 0))

### (60) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

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

Use narrowing to replace COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0))) by

COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))

### (62) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

### (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.

COND1(true, 0, z0) → c(COND2(false, 0, z0))
We considered the (Usable) Rules:

gr(s(z0), 0) → true
and(z0, false) → false
and(false, z0) → false
and(true, true) → true
gr(0, z0) → false
And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = [2]x1
POL(COND2(x1, x2, x3)) = [2]x2
POL(COND3(x1, x2, x3)) = [2]
POL(COND4(x1, x2, x3)) = [2]x2
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = x1
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = x1
POL(p(x1)) = [2]
POL(s(x1)) = [1]
POL(true) = [1]

### (64) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

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

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

COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))

### (66) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

### (67) 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.

COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
We considered the (Usable) Rules:none
And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = [2]x2
POL(COND2(x1, x2, x3)) = [2]x2
POL(COND3(x1, x2, x3)) = x1·x2
POL(COND4(x1, x2, x3)) = [2]x2
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1)) = x1
POL(false) = [2]
POL(gr(x1, x2)) = 0
POL(p(x1)) = 0
POL(s(x1)) = [2]
POL(true) = 0

### (68) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

### (69) 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.

COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
We considered the (Usable) Rules:

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

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = x2
POL(COND2(x1, x2, x3)) = x1·x2
POL(COND3(x1, x2, x3)) = x1·x2
POL(COND4(x1, x2, x3)) = x2
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1)) = x1
POL(false) = [1]
POL(gr(x1, x2)) = [1]
POL(p(x1)) = 0
POL(s(x1)) = [2]
POL(true) = 0

### (70) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

### (71) 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.

COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
We considered the (Usable) Rules:

p(s(z0)) → z0
gr(s(z0), 0) → true
p(0) → 0
gr(0, z0) → false
And the Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = [2]x32
POL(COND2(x1, x2, x3)) = [2]x32
POL(COND3(x1, x2, x3)) = [2]x32
POL(COND4(x1, x2, x3)) = [2]x32 + [2]x1·x3
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1)) = x1
POL(false) = [2]
POL(gr(x1, x2)) = [2]
POL(p(x1)) = x1
POL(s(x1)) = [2] + x1
POL(true) = 0

### (72) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

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

Use narrowing to replace COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1)) by

COND3(false, 0, x0) → c4(COND1(false, 0, x0))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, 0, s(z0)) → c4(COND1(and(false, true), 0, s(z0)))

### (74) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND3(false, 0, x0) → c4(COND1(false, 0, x0))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND3(false, 0, x1) → c4(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(false, x0, s(z0)) → c4(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

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

Removed 1 trailing nodes:

COND3(false, 0, x0) → c4(COND1(false, 0, x0))

### (76) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND3, COND4

Compound Symbols:

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

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

Use narrowing to replace COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1)) by

COND3(false, s(x0), 0) → c4(COND1(and(true, false), s(x0), 0))
COND3(false, s(x0), s(z0)) → c4(COND1(and(true, true), s(x0), s(z0)))

### (78) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

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

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

Use narrowing to replace COND4(true, x0, 0) → c5(COND4(gr(0, 0), x0, 0)) by

COND4(true, x0, 0) → c5(COND4(false, x0, 0))

### (80) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

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

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

Use narrowing to replace COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0)) by

COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))

### (82) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(false, 0, 0) → c4(COND1(and(false, false), 0, 0))
COND3(false, s(z0), 0) → c4(COND1(and(true, false), s(z0), 0))
COND3(false, 0, s(x1)) → c4(COND1(and(false, true), 0, s(x1)))
COND3(false, s(z0), s(x1)) → c4(COND1(and(true, true), s(z0), s(x1)))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, z0))
S tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, 0) → c5(COND4(false, x0, p(0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(COND2(false, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND4(true, x0, 0) → c5(COND4(false, x0, 0))
K tuples:

COND4(true, x0, s(z0)) → c5(COND4(gr(s(z0), 0), x0, z0))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND1(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND3(false, s(z0), x1) → c4(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND3(true, s(z0), x1) → c3(COND3(true, z0, x1))
COND3(true, 0, x0) → c3(COND3(false, 0, x0))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND1(true, s(s(z0)), s(0)) → c(COND2(true, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
Defined Rule Symbols:

gr, p, and

Defined Pair Symbols:

GR, COND1, COND2, COND4, COND3

Compound Symbols:

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

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

Use narrowing to replace COND4(true, x0, 0) → c5(COND4(false, x0, p(0))) by

COND4(true, x0, 0) → c5(COND4(false, x0, 0))

### (84) Obligation:

Complexity Dependency Tuples Problem
Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
and(true, true) → true
and(false, z0) → false
and(z0, false) → false
Tuples:

GR(s(z0), s(z1)) → c9(GR(z0, z1))
COND1(true, 0, z0) → c(COND2(false, 0, z0))
COND1(true, s(z0), 0) → c(COND2(true, s(z0), 0))
COND2(true, s(z0), x1) → c1(COND3(true, s(z0), x1))
COND2(false, x0, 0) → c2(COND4(false, x0, 0))
COND2(false, x0, s(z0)) → c2(COND4(true, x0, s(z0)))
COND4(true, x0, s(z0)) → c5(COND4(true, x0, p(s(z0))))
COND4(false, x0, 0) → c6(COND1(and(gr(x0, 0), false), x0, 0))
COND4(false, x0, s(z0)) → c6(COND1(and(gr(x0, 0), true), x0, s(z0)))
COND4(false, 0, x1) → c6(COND1(and(false, gr(x1, 0)), 0, x1))
COND4(false, s(z0), x1) → c6(COND1(and(true, gr(x1, 0)), s(z0), x1))
COND1(true, s(0), s(z0)) → c(