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

(0) Obligation:

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


The TRS R consists of the following rules:

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

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond1(true, z0, z1, z2) → cond2(gr(z1, z2), z0, z1, z2)
cond2(true, z0, z1, z2) → cond2(gr(z1, z2), z0, p(z1), z2)
cond2(false, z0, z1, z2) → cond1(gr(z0, z2), p(z0), z1, z2)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0

Tuples:

COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2), P(z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2), P(z0))
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

S tuples:

COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2), P(z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2), P(z0))
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

K tuples:none
Defined Rule Symbols:

cond1, cond2, gr, p

Defined Pair Symbols:

COND1, COND2, GR, P

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7

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

Removed 4 trailing nodes:

P(s(z0)) → c7
P(0) → c6
GR(s(z0), 0) → c4
GR(0, z0) → c3

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond1(true, z0, z1, z2) → cond2(gr(z1, z2), z0, z1, z2)
cond2(true, z0, z1, z2) → cond2(gr(z1, z2), z0, p(z1), z2)
cond2(false, z0, z1, z2) → cond1(gr(z0, z2), p(z0), z1, z2)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0

Tuples:

COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2), P(z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2), P(z0))
GR(s(z0), s(z1)) → c5(GR(z0, z1))

S tuples:

COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2), P(z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2), P(z0))
GR(s(z0), s(z1)) → c5(GR(z0, z1))

K tuples:none
Defined Rule Symbols:

cond1, cond2, gr, p

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c5

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

Removed 2 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond1(true, z0, z1, z2) → cond2(gr(z1, z2), z0, z1, z2)
cond2(true, z0, z1, z2) → cond2(gr(z1, z2), z0, p(z1), z2)
cond2(false, z0, z1, z2) → cond1(gr(z0, z2), p(z0), z1, z2)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0

Tuples:

COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))

S tuples:

COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))

K tuples:none
Defined Rule Symbols:

cond1, cond2, gr, p

Defined Pair Symbols:

COND1, GR, COND2

Compound Symbols:

c, c5, c1, c2

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

cond1(true, z0, z1, z2) → cond2(gr(z1, z2), z0, z1, z2)
cond2(true, z0, z1, z2) → cond2(gr(z1, z2), z0, p(z1), z2)
cond2(false, z0, z1, z2) → cond1(gr(z0, z2), p(z0), z1, z2)

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))

S tuples:

COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))

K tuples:none
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND1, GR, COND2

Compound Symbols:

c, c5, c1, c2

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

Use narrowing to replace COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2)) by

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0), GR(0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0), GR(s(z0), 0))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0), GR(0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0), GR(s(z0), 0))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))

K tuples:none
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND2, COND1

Compound Symbols:

c5, c1, c2, c

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

Removed 2 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))

K tuples:none
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND2, COND1

Compound Symbols:

c5, c1, c2, c, c

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

Use narrowing to replace COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2)) by

COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2), GR(0, x2))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0), GR(0, z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0), GR(s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2), GR(0, x2))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0), GR(0, z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0), GR(s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2), GR(0, x2))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0), GR(0, z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0), GR(s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))

K tuples:none
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND2, COND1

Compound Symbols:

c5, c2, c, c, c1

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

Removed 3 trailing tuple parts

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))

K tuples:none
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND2, COND1

Compound Symbols:

c5, c2, c, c, c1, c1

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(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), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))

We considered the (Usable) Rules:

p(0) → 0
p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₃   
POL(COND2(x₁, x₂, x₃, x₄)) = x₃   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))

Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND2, COND1

Compound Symbols:

c5, c2, c, c, c1, c1

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

Use narrowing to replace COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2)) by

COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2), GR(0, x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, 0, x1, z0) → c2(COND1(false, p(0), x1, z0), GR(0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0), GR(s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2), GR(0, x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, 0, x1, z0) → c2(COND1(false, p(0), x1, z0), GR(0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0), GR(s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2), GR(0, x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, 0, x1, z0) → c2(COND1(false, p(0), x1, z0), GR(0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0), GR(s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))

Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c1, c2

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

Removed 1 trailing nodes:

COND2(false, 0, x1, z0) → c2(COND1(false, p(0), x1, z0), GR(0, z0))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2), GR(0, x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0), GR(s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2), GR(0, x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0), GR(s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))

Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c1, c2

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

Removed 2 trailing tuple parts

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))

Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c1, c2, c2

(25) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))

We considered the (Usable) Rules:

p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₂   
POL(COND2(x₁, x₂, x₃, x₄)) = x₂   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))

Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c1, c2, c2

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

Use narrowing to replace COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1))) by

COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))

Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c1, c1, c2, c2, c

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

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

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₄   
POL(COND2(x₁, x₂, x₃, x₄)) = x₄   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = [1]   
POL(s(x₁)) = [1]   
POL(true) = 0   

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))

Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c1, c1, c2, c2, c

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

Use narrowing to replace COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2)) by

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0), GR(s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0), GR(s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))

Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c1, c1, c2, c2, c

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

Removed 1 trailing tuple parts

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))

Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c1, c1, c2, c2, c

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

Use narrowing to replace COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1))) by

COND2(true, x0, s(z0), s(x2)) → c1(COND2(gr(z0, x2), x0, z0, s(x2)), GR(s(z0), s(x2)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(z0), s(x2)) → c1(COND2(gr(z0, x2), x0, z0, s(x2)), GR(s(z0), s(x2)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))

Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c1, c2, c2, c, c1

(37) CdtRuleRemovalProof (UPPER BOUND(ADD(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), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))

We considered the (Usable) Rules:

p(0) → 0
p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(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, x₄)) = x₃   
POL(COND2(x₁, x₂, x₃, x₄)) = x₃   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))

Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c1, c2, c2, c, c1

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

Use narrowing to replace COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2)) by

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

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))

Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c1, c2, c2, c, c1

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

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

COND2(true, x0, 0, x1) → c1(COND2(false, x0, 0, x1))

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))

Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c1, c2, c2, c, c1

(43) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

p(0) → 0

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c1, c2, c2, c, c1

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

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

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

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c2, c2, c, c1, c1

(47) CdtRuleRemovalProof (UPPER BOUND(ADD(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), 0) → c1(COND2(true, x0, z0, 0))

We considered the (Usable) Rules:

p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₃   
POL(COND2(x₁, x₂, x₃, x₄)) = x₃   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c2, c2, c, c1, c1

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

Use narrowing to replace COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2)) by

COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0), GR(s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0), GR(s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c2, c2, c, c1, c1

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

Removed 1 trailing tuple parts

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c2, c2, c, c1, c1

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

Use narrowing to replace COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1))) by

COND2(false, s(z0), x1, s(x2)) → c2(COND1(gr(z0, x2), z0, x1, s(x2)), GR(s(z0), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(COND1(false, p(s(0)), x1, s(z0)), GR(s(0), s(z0)))

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(COND1(false, p(s(0)), x1, s(z0)), GR(s(0), s(z0)))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(x2)) → c2(COND1(gr(z0, x2), z0, x1, s(x2)), GR(s(z0), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(COND1(false, p(s(0)), x1, s(z0)), GR(s(0), s(z0)))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c2, c, c1, c1, c2

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

Removed 1 trailing tuple parts

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(x2)) → c2(COND1(gr(z0, x2), z0, x1, s(x2)), GR(s(z0), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c2, c, c1, c1, c2

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

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

COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))

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

POL(0) = [2]   
POL(COND1(x₁, x₂, x₃, x₄)) = [2]x₄   
POL(COND2(x₁, x₂, x₃, x₄)) = [2]x₄   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = 0   
POL(s(x₁)) = [2]   
POL(true) = 0   

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(x2)) → c2(COND1(gr(z0, x2), z0, x1, s(x2)), GR(s(z0), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c2, c, c1, c1, c2

(59) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))

We considered the (Usable) Rules:

p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₂   
POL(COND2(x₁, x₂, x₃, x₄)) = x₂   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c2, c, c1, c1, c2

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

Use narrowing to replace COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2)) by

COND2(false, 0, x0, z0) → c2(COND1(false, 0, x0, z0))

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, 0, x0, z0) → c2(COND1(false, 0, x0, z0))

S tuples:

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

K tuples:

COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c2, c, c1, c1, c2

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

Removed 1 trailing nodes:

COND2(false, 0, x0, z0) → c2(COND1(false, 0, x0, z0))

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c2, c, c1, c1, c2

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

Use narrowing to replace COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0)) by

COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c1, c2, c2

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

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

COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))

We considered the (Usable) Rules:

p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₂   
POL(COND2(x₁, x₂, x₃, x₄)) = x₂   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c1, c2, c2

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

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

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c1, c2, c2

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

Use narrowing to replace COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1)))) by

COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c1, c2, c2

(73) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₄   
POL(COND2(x₁, x₂, x₃, x₄)) = x₄   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = 0   
POL(s(x₁)) = [1]   
POL(true) = 0   

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c1, c2, c2

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

Use narrowing to replace COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1))) by

COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c1, c2, c2

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

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

COND2(true, x0, s(0), s(x1)) → c1(COND2(false, x0, 0, s(x1)), GR(s(0), s(x1)))

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c1, c2, c2

(79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

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

We considered the (Usable) Rules:

p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₃   
POL(COND2(x₁, x₂, x₃, x₄)) = x₃   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c1, c2, c2

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

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

COND2(true, x0, s(s(x1)), s(0)) → c1(COND2(true, x0, s(x1), s(0)), GR(s(s(x1)), s(0)))

(82) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c1, c2, c2

(83) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))

We considered the (Usable) Rules:

p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₃   
POL(COND2(x₁, x₂, x₃, x₄)) = x₃   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(84) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c1, c2, c2

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

Use narrowing to replace COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1)))) by

COND2(true, x0, s(s(x1)), s(s(x2))) → c1(COND2(gr(x1, x2), x0, s(x1), s(s(x2))), GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))

(86) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(COND2(gr(x1, x2), x0, s(x1), s(s(x2))), GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c2, c1

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

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

COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))

We considered the (Usable) Rules:

p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₃   
POL(COND2(x₁, x₂, x₃, x₄)) = x₃   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(88) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c2, c1

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

Use narrowing to replace COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1))) by

COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(COND1(false, 0, x1, s(z0)), GR(s(0), s(z0)))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))

(90) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(COND1(false, 0, x1, s(z0)), GR(s(0), s(z0)))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c2, c1

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

Removed 1 trailing tuple parts

(92) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c2, c1

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

Use narrowing to replace COND2(false, s(z0), x1, s(x2)) → c2(COND1(gr(z0, x2), z0, x1, s(x2)), GR(s(z0), s(x2))) by

COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(COND1(false, 0, x1, s(z0)), GR(s(0), s(z0)))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))

(94) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(0), x1, s(z0)) → c2(COND1(false, 0, x1, s(z0)), GR(s(0), s(z0)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c2, c1

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

Removed 1 trailing tuple parts

(96) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c2, c1

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

Use narrowing to replace COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0))) by

COND2(false, s(s(x0)), x1, s(0)) → c2(COND1(true, s(x0), x1, s(0)), GR(s(s(x0)), s(0)))

(98) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c2, c1

(99) CdtRuleRemovalProof (UPPER BOUND(ADD(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)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))

We considered the (Usable) Rules:

p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₂   
POL(COND2(x₁, x₂, x₃, x₄)) = x₂   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(100) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c2, c1

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

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

COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(x1)), s(0)) → c1(COND2(true, x0, s(x1), s(0)), GR(s(s(x1)), s(0)))

(102) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c2, c1

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

Use narrowing to replace COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1)))) by

COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(COND1(false, p(s(s(0))), x1, s(s(z0))), GR(s(s(0)), s(s(z0))))

(104) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(COND1(false, p(s(s(0))), x1, s(s(z0))), GR(s(s(0)), s(s(z0))))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(COND1(false, p(s(s(0))), x1, s(s(z0))), GR(s(s(0)), s(s(z0))))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c1, c2

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

Removed 1 trailing tuple parts

(106) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c1, c2

(107) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = [1] + x₄   
POL(COND2(x₁, x₂, x₃, x₄)) = [1] + x₄   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = 0   
POL(s(x₁)) = [1]   
POL(true) = 0   

(108) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c1, c2

(109) CdtRuleRemovalProof (UPPER BOUND(ADD(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)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))

We considered the (Usable) Rules:

p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₂   
POL(COND2(x₁, x₂, x₃, x₄)) = x₂   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(110) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c1, c2

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

Use narrowing to replace COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1))))) by

COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))

(112) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c1, c2

(113) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))

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

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))

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

POL(0) = [1]   
POL(COND1(x₁, x₂, x₃, x₄)) = x₄   
POL(COND2(x₁, x₂, x₃, x₄)) = x₄   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = 0   
POL(s(x₁)) = [1]   
POL(true) = 0   

(114) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c1, c2

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

Use narrowing to replace COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1)))) by

COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))

(116) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c1, c2

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

Use narrowing to replace COND2(true, x0, s(s(x1)), s(s(x2))) → c1(COND2(gr(x1, x2), x0, s(x1), s(s(x2))), GR(s(s(x1)), s(s(x2)))) by

COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))

(118) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c1, c2

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

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

COND2(true, x0, s(s(s(x1))), s(s(0))) → c1(COND2(true, x0, s(s(x1)), s(s(0))), GR(s(s(s(x1))), s(s(0))))

(120) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c1, c2

(121) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))

We considered the (Usable) Rules:

p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₃   
POL(COND2(x₁, x₂, x₃, x₄)) = x₃   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(122) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c1, c2

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

Use narrowing to replace COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1))))) by

COND2(true, x0, s(s(s(x1))), s(s(s(x2)))) → c1(COND2(gr(x1, x2), x0, s(s(x1)), s(s(s(x2)))), GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))

(124) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))

S tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(x1))), s(s(s(x2)))) → c1(COND2(gr(x1, x2), x0, s(s(x1)), s(s(s(x2)))), GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c1, c2

(125) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))

We considered the (Usable) Rules:

p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₃   
POL(COND2(x₁, x₂, x₃, x₄)) = x₃   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(126) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c1, c2

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

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

COND2(true, x0, s(s(0)), s(s(x1))) → c1(COND2(false, x0, s(0), s(s(x1))), GR(s(s(0)), s(s(x1))))

(128) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c1, c2

(129) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

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

We considered the (Usable) Rules:

p(s(z0)) → z0

And the Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₃ + x₄   
POL(COND2(x₁, x₂, x₃, x₄)) = x₃ + x₄   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(130) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))

S tuples:

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

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

GR, COND1, COND2

Compound Symbols:

c5, c, c, c1, c2, c1, c2

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

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

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

(132) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

S tuples:

COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c, c1, c2, c1, c2, c5

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

Removed 1 trailing nodes:

COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))

(134) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

S tuples:

COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c, c1, c2, c1, c2, c5

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

Removed 9 trailing tuple parts

(136) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))

S tuples:

COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

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

Used rewriting to replace COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0)))) by COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))

(138) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))

S tuples:

COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

(139) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))

We considered the (Usable) Rules:

p(s(z0)) → z0

And the Tuples:

COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₂   
POL(COND2(x₁, x₂, x₃, x₄)) = x₂   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(140) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))

S tuples:

COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

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

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

COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(x1))), s(s(0))) → c1(COND2(true, x0, s(s(x1)), s(s(0))), GR(s(s(s(x1))), s(s(0))))

(142) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))

S tuples:

COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

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

Use instantiation to replace COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0)) by

COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))

(144) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))

S tuples:

COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

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

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

COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(x0)), x1, s(0)) → c2(COND1(true, s(x0), x1, s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))

(146) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))

S tuples:

COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))

K tuples:

COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

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

Use instantiation to replace COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2))) by

COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(x0), s(z1), s(0)) → c(GR(s(z1), s(0)))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))

(148) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(x0), s(z1), s(0)) → c(GR(s(z1), s(0)))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))

S tuples:

COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(x0), s(z1), s(0)) → c(GR(s(z1), s(0)))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

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

Removed 1 trailing nodes:

COND1(true, s(x0), s(z1), s(0)) → c(GR(s(z1), s(0)))

(150) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))

S tuples:

COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

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

Used rewriting to replace COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1))))) by COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))

(152) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))

S tuples:

COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

(153) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))

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

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))

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

POL(0) = [1]   
POL(COND1(x₁, x₂, x₃, x₄)) = x₂   
POL(COND2(x₁, x₂, x₃, x₄)) = x₂   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = [1]   
POL(gr(x₁, x₂)) = [1] + x₁   
POL(p(x₁)) = 0   
POL(s(x₁)) = [1] + x₁   
POL(true) = [1]   

(154) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))

S tuples:

COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

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

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

COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(s(x1))), s(s(s(x2)))) → c1(COND2(gr(x1, x2), x0, s(s(x1)), s(s(s(x2)))), GR(s(s(s(x1))), s(s(s(x2)))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(s(x1))), s(s(s(x2)))) → c1(COND2(gr(x1, x2), x0, s(s(x1)), s(s(s(x2)))), GR(s(s(s(x1))), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(x0)), x1, s(0)) → c2(COND1(true, s(x0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))

(156) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))

S tuples:

COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

(157) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

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

We considered the (Usable) Rules:

gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)

And the Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = [1] + x₂   
POL(COND2(x₁, x₂, x₃, x₄)) = x₁ + x₂   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = [1]   
POL(p(x₁)) = 0   
POL(s(x₁)) = [1] + x₁   
POL(true) = [1]   

(158) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

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

Use instantiation to replace COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0)) by

COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(0)))) → c1(COND2(false, x0, 0, s(s(s(0)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))

(160) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(0)))) → c1(COND2(false, x0, 0, s(s(s(0)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(0)))) → c1(COND2(false, x0, 0, s(s(s(0)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

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

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

COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))

(162) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(0)))) → c1(COND2(false, x0, 0, s(s(s(0)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(0)))) → c1(COND2(false, x0, 0, s(s(s(0)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

(163) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

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

(164) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

(165) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))

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

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))

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

POL(0) = [1]   
POL(COND1(x₁, x₂, x₃, x₄)) = [1]   
POL(COND2(x₁, x₂, x₃, x₄)) = [1]   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = [1]   
POL(gr(x₁, x₂)) = [1] + x₁   
POL(p(x₁)) = 0   
POL(s(x₁)) = [1] + x₁   
POL(true) = [1]   

(166) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

(167) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))

We considered the (Usable) Rules:

p(s(z0)) → z0

And the Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = [1] + x₂ + x₃ + x₄   
POL(COND2(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = 0   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(168) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

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

Use instantiation to replace COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0)) by

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

(170) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c2, c, c2, c1, c5

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

Use instantiation to replace COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0)) by

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

(172) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND1, COND2, GR

Compound Symbols:

c, c1, c, c2, c2, c1, c5

(173) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

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

(174) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND2, COND1, GR

Compound Symbols:

c1, c, c, c2, c2, c1, c5

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

Use instantiation to replace COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0)))) by

COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))

(176) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND2, COND1, GR

Compound Symbols:

c1, c, c, c2, c2, c1, c5

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

Use forward instantiation to replace COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0)) by

COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))

(178) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND2, COND1, GR

Compound Symbols:

c1, c, c, c2, c2, c1, c5

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

Use instantiation to replace COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0)))) by

COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))

(180) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND2, COND1, GR

Compound Symbols:

c1, c, c2, c2, c, c1, c5

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

Use instantiation to replace COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2)))) by

COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))

(182) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND2, COND1, GR

Compound Symbols:

c1, c2, c2, c, c, c1, c5

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

Use instantiation to replace COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2))) by

COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(s(x2))))) → c1(GR(s(s(s(s(x1)))), s(s(s(s(x2))))))
COND2(true, x0, s(s(x1)), s(s(0))) → c1(GR(s(s(x1)), s(s(0))))
COND2(true, x0, s(s(x1)), s(s(s(x2)))) → c1(GR(s(s(x1)), s(s(s(x2)))))
COND2(true, x0, s(z1), s(s(s(s(x2))))) → c1(GR(s(z1), s(s(s(s(x2))))))
COND2(true, x0, s(s(x1)), s(0)) → c1(GR(s(s(x1)), s(0)))
COND2(true, x0, s(x1), s(0)) → c1(GR(s(x1), s(0)))
COND2(true, x0, s(s(s(x1))), s(s(s(0)))) → c1(GR(s(s(s(x1))), s(s(s(0)))))
COND2(true, s(x0), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, s(s(x0)), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))

(184) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(s(x2))))) → c1(GR(s(s(s(s(x1)))), s(s(s(s(x2))))))
COND2(true, x0, s(s(x1)), s(s(0))) → c1(GR(s(s(x1)), s(s(0))))
COND2(true, x0, s(s(x1)), s(s(s(x2)))) → c1(GR(s(s(x1)), s(s(s(x2)))))
COND2(true, x0, s(z1), s(s(s(s(x2))))) → c1(GR(s(z1), s(s(s(s(x2))))))
COND2(true, x0, s(s(x1)), s(0)) → c1(GR(s(s(x1)), s(0)))
COND2(true, x0, s(x1), s(0)) → c1(GR(s(x1), s(0)))
COND2(true, x0, s(s(s(x1))), s(s(s(0)))) → c1(GR(s(s(s(x1))), s(s(s(0)))))
COND2(true, s(x0), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, s(s(x0)), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND2, COND1, GR

Compound Symbols:

c1, c2, c2, c, c, c1, c5

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

Removed 2 trailing nodes:

COND2(true, x0, s(s(x1)), s(0)) → c1(GR(s(s(x1)), s(0)))
COND2(true, x0, s(x1), s(0)) → c1(GR(s(x1), s(0)))

(186) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(s(x2))))) → c1(GR(s(s(s(s(x1)))), s(s(s(s(x2))))))
COND2(true, x0, s(s(x1)), s(s(0))) → c1(GR(s(s(x1)), s(s(0))))
COND2(true, x0, s(s(x1)), s(s(s(x2)))) → c1(GR(s(s(x1)), s(s(s(x2)))))
COND2(true, x0, s(z1), s(s(s(s(x2))))) → c1(GR(s(z1), s(s(s(s(x2))))))
COND2(true, x0, s(s(s(x1))), s(s(s(0)))) → c1(GR(s(s(s(x1))), s(s(s(0)))))
COND2(true, s(x0), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, s(s(x0)), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))

S tuples:

COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

K tuples:

COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND2, COND1, GR

Compound Symbols:

c1, c2, c2, c, c, c1, c5

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

Used rewriting to replace COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1)))))) by COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))

(188) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(s(x2))))) → c1(GR(s(s(s(s(x1)))), s(s(s(s(x2))))))
COND2(true, x0, s(s(x1)), s(s(0))) → c1(GR(s(s(x1)), s(s(0))))
COND2(true, x0, s(s(x1)), s(s(s(x2)))) → c1(GR(s(s(x1)), s(s(s(x2)))))
COND2(true, x0, s(z1), s(s(s(s(x2))))) → c1(GR(s(z1), s(s(s(s(x2))))))
COND2(true, x0, s(s(s(x1))), s(s(s(0)))) → c1(GR(s(s(s(x1))), s(s(s(0)))))
COND2(true, s(x0), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, s(s(x0)), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))

S tuples:

GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))

K tuples:

COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND2, COND1, GR

Compound Symbols:

c1, c2, c2, c, c, c1, c5

(189) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))

(190) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(s(x2))))) → c1(GR(s(s(s(s(x1)))), s(s(s(s(x2))))))
COND2(true, x0, s(s(x1)), s(s(0))) → c1(GR(s(s(x1)), s(s(0))))
COND2(true, x0, s(s(x1)), s(s(s(x2)))) → c1(GR(s(s(x1)), s(s(s(x2)))))
COND2(true, x0, s(z1), s(s(s(s(x2))))) → c1(GR(s(z1), s(s(s(s(x2))))))
COND2(true, x0, s(s(s(x1))), s(s(s(0)))) → c1(GR(s(s(s(x1))), s(s(s(0)))))
COND2(true, s(x0), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, s(s(x0)), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))

S tuples:

GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))

K tuples:

COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND2, COND1, GR

Compound Symbols:

c1, c2, c2, c, c, c1, c5

(191) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))

We considered the (Usable) Rules:

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

And the Tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(s(x2))))) → c1(GR(s(s(s(s(x1)))), s(s(s(s(x2))))))
COND2(true, x0, s(s(x1)), s(s(0))) → c1(GR(s(s(x1)), s(s(0))))
COND2(true, x0, s(s(x1)), s(s(s(x2)))) → c1(GR(s(s(x1)), s(s(s(x2)))))
COND2(true, x0, s(z1), s(s(s(s(x2))))) → c1(GR(s(z1), s(s(s(s(x2))))))
COND2(true, x0, s(s(s(x1))), s(s(s(0)))) → c1(GR(s(s(s(x1))), s(s(s(0)))))
COND2(true, s(x0), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, s(s(x0)), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))

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

POL(0) = [1]   
POL(COND1(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(COND2(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(GR(x₁, x₂)) = 0   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = [1]   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = [1]   

(192) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(s(x2))))) → c1(GR(s(s(s(s(x1)))), s(s(s(s(x2))))))
COND2(true, x0, s(s(x1)), s(s(0))) → c1(GR(s(s(x1)), s(s(0))))
COND2(true, x0, s(s(x1)), s(s(s(x2)))) → c1(GR(s(s(x1)), s(s(s(x2)))))
COND2(true, x0, s(z1), s(s(s(s(x2))))) → c1(GR(s(z1), s(s(s(s(x2))))))
COND2(true, x0, s(s(s(x1))), s(s(s(0)))) → c1(GR(s(s(s(x1))), s(s(s(0)))))
COND2(true, s(x0), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, s(s(x0)), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))

S tuples:

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

K tuples:

COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND2, COND1, GR

Compound Symbols:

c1, c2, c2, c, c, c1, c5

(193) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

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

We considered the (Usable) Rules:

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

And the Tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(s(x2))))) → c1(GR(s(s(s(s(x1)))), s(s(s(s(x2))))))
COND2(true, x0, s(s(x1)), s(s(0))) → c1(GR(s(s(x1)), s(s(0))))
COND2(true, x0, s(s(x1)), s(s(s(x2)))) → c1(GR(s(s(x1)), s(s(s(x2)))))
COND2(true, x0, s(z1), s(s(s(s(x2))))) → c1(GR(s(z1), s(s(s(s(x2))))))
COND2(true, x0, s(s(s(x1))), s(s(s(0)))) → c1(GR(s(s(s(x1))), s(s(s(0)))))
COND2(true, s(x0), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, s(s(x0)), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))

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

POL(0) = 0   
POL(COND1(x₁, x₂, x₃, x₄)) = x₄ + x₃·x₄ + [2]x₂·x₄ + [2]x₁·x₄ + x₁² + x₂·x₃   
POL(COND2(x₁, x₂, x₃, x₄)) = [1] + x₃·x₄ + [2]x₂·x₄ + [2]x₁·x₄ + x₂·x₃   
POL(GR(x₁, x₂)) = x₂   
POL(c(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(false) = 0   
POL(gr(x₁, x₂)) = [1]   
POL(p(x₁)) = x₁   
POL(s(x₁)) = [2] + x₁   
POL(true) = [1]   

(194) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(s(x2))))) → c1(GR(s(s(s(s(x1)))), s(s(s(s(x2))))))
COND2(true, x0, s(s(x1)), s(s(0))) → c1(GR(s(s(x1)), s(s(0))))
COND2(true, x0, s(s(x1)), s(s(s(x2)))) → c1(GR(s(s(x1)), s(s(s(x2)))))
COND2(true, x0, s(z1), s(s(s(s(x2))))) → c1(GR(s(z1), s(s(s(s(x2))))))
COND2(true, x0, s(s(s(x1))), s(s(s(0)))) → c1(GR(s(s(s(x1))), s(s(s(0)))))
COND2(true, s(x0), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, s(s(x0)), s(s(s(x1))), s(s(0))) → c1(GR(s(s(s(x1))), s(s(0))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))

S tuples:none
K tuples:

COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, s(s(x0)), 0, s(s(0))) → c(COND2(false, s(s(x0)), 0, s(s(0))))
COND1(true, s(x0), s(z1), s(s(x2))) → c(GR(s(z1), s(s(x2))))
COND1(true, y1, s(z1), s(s(s(x2)))) → c(GR(s(z1), s(s(s(x2)))))
COND1(true, s(s(x0)), s(z1), s(s(0))) → c(GR(s(z1), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, s(s(x0)), s(s(0)), s(s(0))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(s(x0)), s(s(0)), s(s(s(x2)))) → c(COND2(false, s(s(x0)), s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, s(s(x0)), s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, s(s(x0)), s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))

Defined Rule Symbols:

p, gr

Defined Pair Symbols:

COND2, COND1, GR

Compound Symbols:

c1, c2, c2, c, c, c1, c5

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

The set S is empty

(196) BOUNDS(1, 1)