### (0) Obligation:

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

cond(true, x, y) → cond(gr(x, y), p(x), s(y))
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1), P(z0))
GR(0, z0) → c1
GR(s(z0), 0) → c2
GR(s(z0), s(z1)) → c3(GR(z0, z1))
P(0) → c4
P(s(z0)) → c5
S tuples:

COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1), P(z0))
GR(0, z0) → c1
GR(s(z0), 0) → c2
GR(s(z0), s(z1)) → c3(GR(z0, z1))
P(0) → c4
P(s(z0)) → c5
K tuples:none
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR, P

Compound Symbols:

c, c1, c2, c3, c4, c5

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

Removed 4 trailing nodes:

GR(0, z0) → c1
P(s(z0)) → c5
GR(s(z0), 0) → c2
P(0) → c4

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1), P(z0))
GR(s(z0), s(z1)) → c3(GR(z0, z1))
S tuples:

COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1), P(z0))
GR(s(z0), s(z1)) → c3(GR(z0, z1))
K tuples:none
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c3

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

Removed 1 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1))
K tuples:none
Defined Rule Symbols:

cond, gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

cond(true, z0, z1) → cond(gr(z0, z1), p(z0), s(z1))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, z0, z1) → c(COND(gr(z0, z1), p(z0), s(z1)), GR(z0, z1))
K tuples:none
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c

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

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

COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)), GR(0, x1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, 0, z0) → c(COND(false, p(0), s(z0)), GR(0, z0))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)), GR(s(z0), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(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)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)), GR(0, x1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, 0, z0) → c(COND(false, p(0), s(z0)), GR(0, z0))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)), GR(s(z0), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)), GR(0, x1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, 0, z0) → c(COND(false, p(0), s(z0)), GR(0, z0))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)), GR(s(z0), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c

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

Removed 1 trailing nodes:

COND(true, 0, z0) → c(COND(false, p(0), s(z0)), GR(0, z0))

### (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)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)), GR(0, x1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)), GR(s(z0), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)), GR(0, x1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)), GR(s(z0), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c

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

Removed 2 trailing tuple parts

### (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)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
K tuples:none
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))

### (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)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

### (17) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

p(0) → 0

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
We considered the (Usable) Rules:

p(s(z0)) → z0
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [4]
POL(COND(x1, x2, x3)) = x2
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(false) = [5]
POL(gr(x1, x2)) = [3] + x2
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = [3]

### (20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

Use narrowing to replace COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1)) by

COND(true, s(z0), 0) → c(COND(true, z0, s(0)), GR(s(z0), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), 0) → c(COND(true, z0, s(0)), GR(s(z0), 0))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), x1) → c(COND(gr(s(z0), x1), z0, s(x1)), GR(s(z0), x1))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

### (23) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

COND(true, s(z0), 0) → c(COND(true, z0, s(0)), GR(s(z0), 0))

### (24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), p(s(z0)), s(s(z1))), GR(s(z0), s(z1)))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

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

COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(COND(false, p(s(0)), s(s(z0))), GR(s(0), s(z0)))

### (26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(COND(false, p(s(0)), s(s(z0))), GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(COND(false, p(s(0)), s(s(z0))), GR(s(0), s(z0)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

Removed 1 trailing tuple parts

### (28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, p(s(s(z0))), s(s(0))), GR(s(s(z0)), s(0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c

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

Split RHS of tuples not part of any SCC

### (30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

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

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

### (32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

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

COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
We considered the (Usable) Rules:none
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [4]
POL(COND(x1, x2, x3)) = [1]
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c3(x1)) = x1
POL(false) = [5]
POL(gr(x1, x2)) = [3] + [4]x1 + [2]x2
POL(p(x1)) = 0
POL(s(x1)) = 0
POL(true) = [3]

### (34) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

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

COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
We considered the (Usable) Rules:

p(s(z0)) → z0
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND(x1, x2, x3)) = x2
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c3(x1)) = x1
POL(false) = [3]
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = 0

### (36) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

Use narrowing to replace COND(true, 0, x1) → c(COND(gr(0, x1), 0, s(x1))) by

COND(true, 0, z0) → c(COND(false, 0, s(z0)))

### (38) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
COND(true, 0, z0) → c(COND(false, 0, s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, 0, z0) → c(COND(false, 0, s(z0)))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

Removed 1 trailing nodes:

COND(true, 0, z0) → c(COND(false, 0, s(z0)))

### (40) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

Use narrowing to replace COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0))) by

COND(true, s(z0), 0) → c(COND(true, z0, s(0)))

### (42) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
COND(true, s(z0), 0) → c(COND(true, z0, s(0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(z0), 0) → c(COND(true, p(s(z0)), s(0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

### (43) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

COND(true, s(z0), 0) → c(COND(true, z0, s(0)))
COND(true, s(s(z0)), s(0)) → c1(GR(s(s(z0)), s(0)))

### (44) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

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

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

### (46) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(s(0))), GR(s(s(z0)), s(0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(COND(false, 0, s(s(z0))), GR(s(0), s(z0)))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

### (47) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(s(0))), GR(s(s(z0)), s(0)))

### (48) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(COND(false, 0, s(s(z0))), GR(s(0), s(z0)))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

Removed 1 trailing tuple parts

### (50) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

Use narrowing to replace COND(true, s(z0), s(x1)) → c(COND(gr(z0, x1), z0, s(s(x1))), GR(s(z0), s(x1))) by

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

### (52) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(s(0))), GR(s(s(z0)), s(0)))
COND(true, s(0), s(z0)) → c(COND(false, 0, s(s(z0))), GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(z0), s(z1)) → c(COND(gr(z0, z1), z0, s(s(z1))), GR(s(z0), s(z1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

### (53) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

COND(true, s(s(z0)), s(0)) → c(COND(true, s(z0), s(s(0))), GR(s(s(z0)), s(0)))

### (54) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(0), s(z0)) → c(COND(false, 0, s(s(z0))), GR(s(0), s(z0)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

Removed 1 trailing tuple parts

### (56) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), p(s(s(z0))), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c1

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

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

COND(true, s(s(x0)), s(s(x1))) → c(COND(gr(x0, x1), s(x0), s(s(s(x1)))), GR(s(s(x0)), s(s(x1))))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, p(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(COND(false, p(s(s(0))), s(s(s(z0)))), GR(s(s(0)), s(s(z0))))

### (58) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, p(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(COND(false, p(s(s(0))), s(s(s(z0)))), GR(s(s(0)), s(s(z0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(x0)), s(s(x1))) → c(COND(gr(x0, x1), s(x0), s(s(s(x1)))), GR(s(s(x0)), s(s(x1))))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, p(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(COND(false, p(s(s(0))), s(s(s(z0)))), GR(s(s(0)), s(s(z0))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c1, c

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

Removed 1 trailing tuple parts

### (60) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, p(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(x0)), s(s(x1))) → c(COND(gr(x0, x1), s(x0), s(s(s(x1)))), GR(s(s(x0)), s(s(x1))))
COND(true, s(s(s(z0))), s(s(0))) → c(COND(true, p(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c1, c

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

Split RHS of tuples not part of any SCC

### (62) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(x0)), s(s(x1))) → c(COND(gr(x0, x1), s(x0), s(s(s(x1)))), GR(s(s(x0)), s(s(x1))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c1, c, c2

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

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

COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))

### (64) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(x0)), s(s(x1))) → c(COND(gr(x0, x1), s(x0), s(s(s(x1)))), GR(s(s(x0)), s(s(x1))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c1, c, c2

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

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

COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
We considered the (Usable) Rules:none
And the Tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND(x1, x2, x3)) = [1] + [3]x3
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(false) = [3]
POL(gr(x1, x2)) = [2] + [2]x1
POL(p(x1)) = [2] + [3]x1
POL(s(x1)) = 0
POL(true) = [4]

### (66) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(x0)), s(s(x1))) → c(COND(gr(x0, x1), s(x0), s(s(s(x1)))), GR(s(s(x0)), s(s(x1))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c1, c, c2

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

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(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)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND(x1, x2, x3)) = [1] + x2
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(false) = [3]
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [2] + x1
POL(true) = 0

### (68) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c1, c, c2

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

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

COND(true, s(s(x0)), s(0)) → c1(COND(true, s(x0), s(s(0))))

### (70) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(x0)), s(0)) → c1(COND(true, s(x0), s(s(0))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
K tuples:

COND(true, s(s(z0)), s(0)) → c1(COND(true, p(s(s(z0))), s(s(0))))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c2, c1

### (71) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

COND(true, s(s(x0)), s(0)) → c1(COND(true, s(x0), s(s(0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(GR(s(s(s(z0))), s(s(0))))

### (72) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
S tuples:

GR(s(z0), s(z1)) → c3(GR(z0, z1))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
K tuples:

COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

GR, COND

Compound Symbols:

c3, c, c, c2

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

Use forward instantiation to replace GR(s(z0), s(z1)) → c3(GR(z0, z1)) by

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))

### (74) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
S tuples:

COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
K tuples:

COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c2, c3

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

Removed 1 trailing nodes:

COND(true, s(0), s(z0)) → c(GR(s(0), s(z0)))

### (76) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
S tuples:

COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
K tuples:

COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c2, c3

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

Use forward instantiation to replace COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1))) by

COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))

### (78) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
S tuples:

COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
K tuples:

COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c2, c3

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

Use instantiation to replace COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1)))) by

COND(true, s(s(z0)), s(s(s(x1)))) → c(COND(gr(z0, s(x1)), s(z0), s(s(s(s(x1))))), GR(s(s(z0)), s(s(s(x1)))))
COND(true, s(s(z0)), s(s(s(s(x1))))) → c(COND(gr(z0, s(s(x1))), s(z0), s(s(s(s(s(x1)))))), GR(s(s(z0)), s(s(s(s(x1))))))
COND(true, s(s(z0)), s(s(s(0)))) → c(COND(gr(z0, s(0)), s(z0), s(s(s(s(0))))), GR(s(s(z0)), s(s(s(0)))))

### (80) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND(true, s(s(z0)), s(s(s(x1)))) → c(COND(gr(z0, s(x1)), s(z0), s(s(s(s(x1))))), GR(s(s(z0)), s(s(s(x1)))))
COND(true, s(s(z0)), s(s(s(s(x1))))) → c(COND(gr(z0, s(s(x1))), s(z0), s(s(s(s(s(x1)))))), GR(s(s(z0)), s(s(s(s(x1))))))
COND(true, s(s(z0)), s(s(s(0)))) → c(COND(gr(z0, s(0)), s(z0), s(s(s(s(0))))), GR(s(s(z0)), s(s(s(0)))))
S tuples:

COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
K tuples:

COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(s(x1)))) → c(COND(gr(z0, s(x1)), s(z0), s(s(s(s(x1))))), GR(s(s(z0)), s(s(s(x1)))))
COND(true, s(s(z0)), s(s(s(s(x1))))) → c(COND(gr(z0, s(s(x1))), s(z0), s(s(s(s(s(x1)))))), GR(s(s(z0)), s(s(s(s(x1))))))
COND(true, s(s(z0)), s(s(s(0)))) → c(COND(gr(z0, s(0)), s(z0), s(s(s(s(0))))), GR(s(s(z0)), s(s(s(0)))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c2, c3

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

Use forward instantiation to replace COND(true, s(x0), s(x1)) → c(GR(s(x0), s(x1))) by

COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))

### (82) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND(true, s(s(z0)), s(s(s(x1)))) → c(COND(gr(z0, s(x1)), s(z0), s(s(s(s(x1))))), GR(s(s(z0)), s(s(s(x1)))))
COND(true, s(s(z0)), s(s(s(s(x1))))) → c(COND(gr(z0, s(s(x1))), s(z0), s(s(s(s(s(x1)))))), GR(s(s(z0)), s(s(s(s(x1))))))
COND(true, s(s(z0)), s(s(s(0)))) → c(COND(gr(z0, s(0)), s(z0), s(s(s(s(0))))), GR(s(s(z0)), s(s(s(0)))))
S tuples:

COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
K tuples:

COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(s(x1)))) → c(COND(gr(z0, s(x1)), s(z0), s(s(s(s(x1))))), GR(s(s(z0)), s(s(s(x1)))))
COND(true, s(s(z0)), s(s(s(s(x1))))) → c(COND(gr(z0, s(s(x1))), s(z0), s(s(s(s(s(x1)))))), GR(s(s(z0)), s(s(s(s(x1))))))
COND(true, s(s(z0)), s(s(s(0)))) → c(COND(gr(z0, s(0)), s(z0), s(s(s(s(0))))), GR(s(s(z0)), s(s(s(0)))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c2, c3

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

Used rewriting to replace COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), p(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1))))) by COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))

### (84) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND(true, s(s(z0)), s(s(s(x1)))) → c(COND(gr(z0, s(x1)), s(z0), s(s(s(s(x1))))), GR(s(s(z0)), s(s(s(x1)))))
COND(true, s(s(z0)), s(s(s(s(x1))))) → c(COND(gr(z0, s(s(x1))), s(z0), s(s(s(s(s(x1)))))), GR(s(s(z0)), s(s(s(s(x1))))))
COND(true, s(s(z0)), s(s(s(0)))) → c(COND(gr(z0, s(0)), s(z0), s(s(s(s(0))))), GR(s(s(z0)), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
S tuples:

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
K tuples:

COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(s(x1)))) → c(COND(gr(z0, s(x1)), s(z0), s(s(s(s(x1))))), GR(s(s(z0)), s(s(s(x1)))))
COND(true, s(s(z0)), s(s(s(s(x1))))) → c(COND(gr(z0, s(s(x1))), s(z0), s(s(s(s(s(x1)))))), GR(s(s(z0)), s(s(s(s(x1))))))
COND(true, s(s(z0)), s(s(s(0)))) → c(COND(gr(z0, s(0)), s(z0), s(s(s(s(0))))), GR(s(s(z0)), s(s(s(0)))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c2, c3

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

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

COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(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:

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND(true, s(s(z0)), s(s(s(x1)))) → c(COND(gr(z0, s(x1)), s(z0), s(s(s(s(x1))))), GR(s(s(z0)), s(s(s(x1)))))
COND(true, s(s(z0)), s(s(s(s(x1))))) → c(COND(gr(z0, s(s(x1))), s(z0), s(s(s(s(s(x1)))))), GR(s(s(z0)), s(s(s(s(x1))))))
COND(true, s(s(z0)), s(s(s(0)))) → c(COND(gr(z0, s(0)), s(z0), s(s(s(s(0))))), GR(s(s(z0)), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [4]
POL(COND(x1, x2, x3)) = [2]x2
POL(GR(x1, x2)) = [1]
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(false) = [4]
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = 0

### (86) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND(true, s(s(z0)), s(s(s(x1)))) → c(COND(gr(z0, s(x1)), s(z0), s(s(s(s(x1))))), GR(s(s(z0)), s(s(s(x1)))))
COND(true, s(s(z0)), s(s(s(s(x1))))) → c(COND(gr(z0, s(s(x1))), s(z0), s(s(s(s(s(x1)))))), GR(s(s(z0)), s(s(s(s(x1))))))
COND(true, s(s(z0)), s(s(s(0)))) → c(COND(gr(z0, s(0)), s(z0), s(s(s(s(0))))), GR(s(s(z0)), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
S tuples:

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
K tuples:

COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(s(x1)))) → c(COND(gr(z0, s(x1)), s(z0), s(s(s(s(x1))))), GR(s(s(z0)), s(s(s(x1)))))
COND(true, s(s(z0)), s(s(s(s(x1))))) → c(COND(gr(z0, s(s(x1))), s(z0), s(s(s(s(s(x1)))))), GR(s(s(z0)), s(s(s(s(x1))))))
COND(true, s(s(z0)), s(s(s(0)))) → c(COND(gr(z0, s(0)), s(z0), s(s(s(s(0))))), GR(s(s(z0)), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c2, c3

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

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

GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
We considered the (Usable) Rules:

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND(true, s(s(z0)), s(s(s(x1)))) → c(COND(gr(z0, s(x1)), s(z0), s(s(s(s(x1))))), GR(s(s(z0)), s(s(s(x1)))))
COND(true, s(s(z0)), s(s(s(s(x1))))) → c(COND(gr(z0, s(s(x1))), s(z0), s(s(s(s(s(x1)))))), GR(s(s(z0)), s(s(s(s(x1))))))
COND(true, s(s(z0)), s(s(s(0)))) → c(COND(gr(z0, s(0)), s(z0), s(s(s(s(0))))), GR(s(s(z0)), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND(x1, x2, x3)) = [2]x1 + x2 + x22
POL(GR(x1, x2)) = [1] + x1
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = x1
POL(p(x1)) = x1
POL(s(x1)) = [2] + x1
POL(true) = 0

### (88) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, s(s(z0)), s(s(z1))) → c(COND(gr(z0, z1), s(z0), s(s(s(z1)))), GR(s(s(z0)), s(s(z1))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
COND(true, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND(true, s(s(z0)), s(s(s(x1)))) → c(COND(gr(z0, s(x1)), s(z0), s(s(s(s(x1))))), GR(s(s(z0)), s(s(s(x1)))))
COND(true, s(s(z0)), s(s(s(s(x1))))) → c(COND(gr(z0, s(s(x1))), s(z0), s(s(s(s(s(x1)))))), GR(s(s(z0)), s(s(s(s(x1))))))
COND(true, s(s(z0)), s(s(s(0)))) → c(COND(gr(z0, s(0)), s(z0), s(s(s(s(0))))), GR(s(s(z0)), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
S tuples:none
K tuples:

COND(true, s(s(s(z0))), s(s(0))) → c2(COND(true, p(s(s(s(z0)))), s(s(s(0)))))
COND(true, s(s(0)), s(s(z0))) → c(GR(s(s(0)), s(s(z0))))
COND(true, s(s(z0)), s(s(s(x1)))) → c(COND(gr(z0, s(x1)), s(z0), s(s(s(s(x1))))), GR(s(s(z0)), s(s(s(x1)))))
COND(true, s(s(z0)), s(s(s(s(x1))))) → c(COND(gr(z0, s(s(x1))), s(z0), s(s(s(s(s(x1)))))), GR(s(s(z0)), s(s(s(s(x1))))))
COND(true, s(s(z0)), s(s(s(0)))) → c(COND(gr(z0, s(0)), s(z0), s(s(s(s(0))))), GR(s(s(z0)), s(s(s(0)))))
COND(true, s(s(s(z0))), s(s(s(z1)))) → c(COND(gr(z0, z1), s(s(z0)), s(s(s(s(z1))))), GR(s(s(s(z0))), s(s(s(z1)))))
GR(s(s(y0)), s(s(y1))) → c3(GR(s(y0), s(y1)))
Defined Rule Symbols:

gr, p

Defined Pair Symbols:

COND, GR

Compound Symbols:

c, c, c2, c3

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

The set S is empty