(0) Obligation:

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

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

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

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

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

cond1, cond2, gr, p, eq, and

Defined Pair Symbols:

COND1, COND2, GR, P, EQ, AND

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 10 trailing nodes:

EQ(0, s(z0)) → c10
AND(z0, false) → c14
GR(0, z0) → c3
GR(s(z0), 0) → c4
EQ(s(z0), 0) → c9
AND(true, true) → c12
P(0) → c6
AND(false, z0) → c13
P(s(z0)) → c7
EQ(0, 0) → c8

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1), GR(z1, 0))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)), GR(z1, 0), P(z0), P(z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), AND(eq(z0, z1), gr(z0, 0)), EQ(z0, z1), GR(z0, 0))
GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
S tuples:

COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1), GR(z1, 0))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)), GR(z1, 0), P(z0), P(z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), AND(eq(z0, z1), gr(z0, 0)), EQ(z0, z1), GR(z0, 0))
GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
K tuples:none
Defined Rule Symbols:

cond1, cond2, gr, p, eq, and

Defined Pair Symbols:

COND1, COND2, GR, EQ

Compound Symbols:

c, c1, c2, c5, c11

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

Removed 6 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
K tuples:none
Defined Rule Symbols:

cond1, cond2, gr, p, eq, and

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

cond1(true, z0, z1) → cond2(gr(z1, 0), z0, z1)
cond2(true, z0, z1) → cond2(gr(z1, 0), p(z0), p(z1))
cond2(false, z0, z1) → cond1(and(eq(z0, z1), gr(z0, 0)), z0, z1)
gr(s(z0), s(z1)) → gr(z0, z1)

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
K tuples:none
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2

(9) 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.

GR(s(z0), s(z1)) → c5(GR(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = 0
POL(COND2(x1, x2, x3)) = 0
POL(EQ(x1, x2)) = 0
POL(GR(x1, x2)) = [4]x1 + [4]x2
POL(and(x1, x2)) = 0
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(eq(x1, x2)) = [2] + [4]x1
POL(false) = [2]
POL(gr(x1, x2)) = [5]x1 + [2]x2
POL(p(x1)) = [3]
POL(s(x1)) = [4] + x1
POL(true) = 0

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2

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

Use narrowing to replace COND1(true, z0, z1) → c(COND2(gr(z1, 0), z0, z1)) by

COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(true, z0, z1) → c1(COND2(gr(z1, 0), p(z0), p(z1)))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND2, COND1

Compound Symbols:

c5, c11, c1, c2, c

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

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

COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND2, COND1

Compound Symbols:

c5, c11, c2, c, c1

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

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

COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
We considered the (Usable) Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(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]x2
POL(COND2(x1, x2, x3)) = [4]x2
POL(EQ(x1, x2)) = 0
POL(GR(x1, x2)) = [3]x1 + [4]x2
POL(and(x1, x2)) = 0
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(eq(x1, x2)) = 0
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [2] + x1
POL(true) = [1]

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND2, COND1

Compound Symbols:

c5, c11, c2, c, c1

(17) 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.

COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
We considered the (Usable) Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = [2]x3
POL(COND2(x1, x2, x3)) = [2]x3
POL(EQ(x1, x2)) = 0
POL(GR(x1, x2)) = [5]x1 + [4]x2
POL(and(x1, x2)) = 0
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(eq(x1, x2)) = [2]x2
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = 0

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND2, COND1

Compound Symbols:

c5, c11, c2, c, c1

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

Use narrowing to replace COND2(false, z0, z1) → c2(COND1(and(eq(z0, z1), gr(z0, 0)), z0, z1), EQ(z0, z1)) by

COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1), EQ(0, x1))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0), EQ(0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0), EQ(s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)), EQ(0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1), EQ(0, x1))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0), EQ(0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0), EQ(s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)), EQ(0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1), EQ(0, x1))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0), EQ(0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0), EQ(s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)), EQ(0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2

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

Removed 4 trailing tuple parts

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

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

COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(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)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = [3] + [2]x1
POL(COND2(x1, x2, x3)) = [5]
POL(EQ(x1, x2)) = 0
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = x2
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1)) = x1
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(eq(x1, x2)) = 0
POL(false) = 0
POL(gr(x1, x2)) = x1
POL(p(x1)) = 0
POL(s(x1)) = [1]
POL(true) = [1]

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

(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.

COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
We considered the (Usable) Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
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(EQ(x1, x2)) = 0
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = x1
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1)) = x1
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(eq(x1, x2)) = [1]
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = 0
POL(s(x1)) = 0
POL(true) = [1]

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, x0, 0) → c1(COND2(gr(0, 0), p(x0), 0)) by

COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

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

COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
We considered the (Usable) Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
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(EQ(x1, x2)) = 0
POL(GR(x1, x2)) = [4]x1 + x2
POL(and(x1, x2)) = 0
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1)) = x1
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(eq(x1, x2)) = [4]x1
POL(false) = 0
POL(gr(x1, x2)) = x2
POL(p(x1)) = x1
POL(s(x1)) = [1] + x1
POL(true) = [2]

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0)) by

COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, 0, x1) → c1(COND2(gr(x1, 0), 0, p(x1))) by

COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, 0, s(z0)) → c1(COND2(gr(s(z0), 0), 0, z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(z0)) → c1(COND2(gr(s(z0), 0), 0, z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

(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.

COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
We considered the (Usable) Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = x3
POL(COND2(x1, x2, x3)) = x3
POL(EQ(x1, x2)) = 0
POL(GR(x1, x2)) = [2]x1 + [3]x2
POL(and(x1, x2)) = 0
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1)) = x1
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(eq(x1, x2)) = [3] + [4]x2
POL(false) = 0
POL(gr(x1, x2)) = [3]x2
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = 0

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1))) by

COND2(true, s(x0), 0) → c1(COND2(gr(0, 0), x0, 0))
COND2(true, s(x0), s(z0)) → c1(COND2(gr(s(z0), 0), x0, z0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, x0, 0) → c1(COND2(false, p(x0), p(0))) by

COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, s(z0), 0) → c1(COND2(false, z0, p(0)))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(z0), 0) → c1(COND2(false, z0, p(0)))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

(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.

COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
We considered the (Usable) Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(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)) = x2
POL(COND2(x1, x2, x3)) = x2
POL(EQ(x1, x2)) = 0
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = 0
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1)) = x1
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(eq(x1, x2)) = 0
POL(false) = 0
POL(gr(x1, x2)) = [2]x2
POL(p(x1)) = x1
POL(s(x1)) = [2] + x1
POL(true) = 0

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0))))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), p(s(z0)))) by

COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, s(x1)) → c1(COND2(true, 0, p(s(x1))))
COND2(true, s(z0), s(x1)) → c1(COND2(true, z0, p(s(x1))))

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, s(z0), s(x1)) → c1(COND2(true, z0, p(s(x1))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c2, c1

(45) 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.

COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
We considered the (Usable) Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(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)) = x2
POL(COND2(x1, x2, x3)) = x2
POL(EQ(x1, x2)) = 0
POL(GR(x1, x2)) = [5]x2
POL(and(x1, x2)) = 0
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1)) = x1
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(eq(x1, x2)) = [2] + [3]x2
POL(false) = 0
POL(gr(x1, x2)) = [1] + [2]x2
POL(p(x1)) = x1
POL(s(x1)) = [1] + x1
POL(true) = 0

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c2, c1

(47) 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.

COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
We considered the (Usable) Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(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)) = [2]x3
POL(COND2(x1, x2, x3)) = [2]x3
POL(EQ(x1, x2)) = 0
POL(GR(x1, x2)) = [4]x1 + [3]x2
POL(and(x1, x2)) = 0
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1)) = x1
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(eq(x1, x2)) = 0
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [2] + x1
POL(true) = 0

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c2, c1

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

Use narrowing to replace COND2(false, s(z0), x1) → c2(COND1(and(eq(s(z0), x1), true), s(z0), x1), EQ(s(z0), x1)) by

COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0), EQ(s(z0), 0))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0), EQ(s(z0), 0))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0), EQ(s(z0), 0))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c2, c1

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

Removed 1 trailing tuple parts

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c2, c1

(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.

COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
We considered the (Usable) Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = x1 + x2
POL(COND2(x1, x2, x3)) = [1]
POL(EQ(x1, x2)) = 0
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = x1
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1)) = x1
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(eq(x1, x2)) = [1]
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = 0
POL(s(x1)) = 0
POL(true) = [1]

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c2, c1

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

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

COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c2, c1

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

Use narrowing to replace COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), gr(s(z0), 0)), s(z0), s(z1)), EQ(s(z0), s(z1))) by

COND2(false, s(z0), s(x1)) → c2(COND1(and(eq(z0, x1), true), s(z0), s(x1)), EQ(s(z0), s(x1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c1, c2

(59) 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.

COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
We considered the (Usable) Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND1(x1, x2, x3)) = [2] + [2]x1
POL(COND2(x1, x2, x3)) = [4]
POL(EQ(x1, x2)) = 0
POL(GR(x1, x2)) = 0
POL(and(x1, x2)) = x1
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1)) = x1
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(eq(x1, x2)) = [1]
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = [2]
POL(s(x1)) = 0
POL(true) = [1]

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c1, c2

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

Use narrowing to replace COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1)) by

COND2(false, 0, x0) → c2(COND1(false, 0, x0))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, x0) → c2(COND1(false, 0, x0))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), x1) → c1(COND2(gr(x1, 0), z0, p(x1)))
COND2(true, x0, s(z0)) → c1(COND2(gr(s(z0), 0), p(x0), z0))
COND2(false, 0, x1) → c2(COND1(and(eq(0, x1), false), 0, x1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c1, c2

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

Removed 1 trailing nodes:

COND2(false, 0, x0) → c2(COND1(false, 0, x0))

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c1, c2

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

Use narrowing to replace COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0)) by

COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c1, c2

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

Use narrowing to replace COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0)) by

COND2(false, s(x0), 0) → c2(COND1(false, s(x0), 0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(false, s(x0), 0) → c2(COND1(false, s(x0), 0))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, 0, 0) → c2(COND1(and(true, gr(0, 0)), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(false, s(z0), 0) → c2(COND1(and(false, gr(s(z0), 0)), s(z0), 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c1, c2

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

Removed 1 trailing nodes:

COND2(false, s(x0), 0) → c2(COND1(false, s(x0), 0))

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c2, c1, c2

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

Use narrowing to replace COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0))) by

COND2(false, 0, s(x0)) → c2(COND1(false, 0, s(x0)))
COND2(false, 0, s(x0)) → c2(COND1(and(false, false), 0, s(x0)))

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(false, 0, s(x0)) → c2(COND1(false, 0, s(x0)))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, 0, s(z0)) → c2(COND1(and(false, gr(0, 0)), 0, s(z0)))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Removed 1 trailing nodes:

COND2(false, 0, s(x0)) → c2(COND1(false, 0, s(x0)))

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

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

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

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

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

COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, 0) → c1(COND2(gr(0, 0), 0, 0))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

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

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

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, 0, 0) → c1(COND2(false, 0, p(0))) by

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

(82) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, s(x0), 0) → c1(COND2(gr(0, 0), x0, 0)) by

COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))

(84) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
Defined Rule Symbols:

gr, p, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

(85) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

gr(0, z0) → false

(86) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(false, 0, 0) → c2(COND1(and(true, false), 0, 0))
COND2(false, 0, s(z0)) → c2(COND1(and(false, false), 0, s(z0)))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
COND2(true, s(x0), 0) → c1(COND2(false, x0, 0))
S tuples:

EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, x0, 0) → c1(COND2(false, p(x0), 0))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)), s(s(z1))), EQ(s(s(z0)), s(s(z1))))
COND2(true, 0, 0) → c1(COND2(false, 0, 0))
K tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, s(z0), 0) → c1(COND2(gr(0, 0), z0, 0))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
Defined Rule Symbols:

p, gr, and, eq

Defined Pair Symbols:

GR, EQ, COND1, COND2

Compound Symbols:

c5, c11, c, c1, c2, c2

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

Use narrowing to replace COND2(true, x0, 0) → c1(COND2(false, p(x0), 0)) by

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

(88) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
EQ(s(z0), s(z1)) → c11(EQ(z0, z1))
COND1(true, x0, 0) → c(COND2(false, x0, 0))
COND1(true, x0, s(z0)) → c(COND2(true, x0, s(z0)))
COND2(true, 0, s(x1)) → c1(COND2(gr(s(x1), 0), 0, x1))
COND2(true, s(z0), s(x1)) → c1(COND2(gr(s(x1), 0), z0, x1))
COND2(true, x0, s(z0)) → c1(COND2(true, p(x0), z0))
COND2(true, 0, 0) → c1(COND2(false, 0, p(0)))
COND2(true, 0, s(z0)) → c1(COND2(true, 0, p(s(z0))))
COND2(true, s(x0), 0) → c1(COND2(false, x0, p(0)))
COND2(true, s(x0), s(z0)) → c1(COND2(true, x0, p(s(z0))))
COND2(false, s(z0), s(z1)) → c2(COND1(and(eq(z0, z1), true), s(z0), s(z1)), EQ(s(z0), s(z1)))
COND2(false, s(z0), 0) → c2(COND1(and(false, true), s(z0), 0))
COND2(false, s(0), s(0)) → c2(COND1(and(true, gr(s(0), 0)), s(0), s(0)), EQ(s(0), s(0)))
COND2(false, s(s(z0)), s(0)) → c2(COND1(and(false, gr(s(s(z0)), 0)), s(s(z0)), s(0)), EQ(s(s(z0)), s(0)))
COND2(false, s(0), s(s(z0))) → c2(COND1(and(false, gr(s(0), 0)), s(0), s(s(z0))), EQ(s(0), s(s(z0))))
COND2(false, s(s(z0)), s(s(z1))) → c2(COND1(and(eq(z0, z1), gr(s(s(z0)), 0)), s(s(z0)),