### (0) Obligation:

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

cond(true, x) → cond(odd(x), p(p(p(x))))
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
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) → cond(odd(z0), p(p(p(z0))))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

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

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

cond, odd, p

Defined Pair Symbols:

COND, ODD, P

Compound Symbols:

c, c1, c2, c3, c4, c5

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

Removed 4 trailing nodes:

P(0) → c4
ODD(0) → c1
ODD(s(0)) → c2
P(s(z0)) → c5

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

COND(true, z0) → c(COND(odd(z0), p(p(p(z0)))), ODD(z0), P(p(p(z0))), P(p(z0)), P(z0))
ODD(s(s(z0))) → c3(ODD(z0))
S tuples:

COND(true, z0) → c(COND(odd(z0), p(p(p(z0)))), ODD(z0), P(p(p(z0))), P(p(z0)), P(z0))
ODD(s(s(z0))) → c3(ODD(z0))
K tuples:none
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c3

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

Removed 3 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, z0) → c(COND(odd(z0), p(p(p(z0)))), ODD(z0))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, z0) → c(COND(odd(z0), p(p(p(z0)))), ODD(z0))
K tuples:none
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

cond(true, z0) → cond(odd(z0), p(p(p(z0))))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, z0) → c(COND(odd(z0), p(p(p(z0)))), ODD(z0))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, z0) → c(COND(odd(z0), p(p(p(z0)))), ODD(z0))
K tuples:none
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c

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

Use narrowing to replace COND(true, z0) → c(COND(odd(z0), p(p(p(z0)))), ODD(z0)) by

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

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), p(p(0))), ODD(0))
COND(true, s(z0)) → c(COND(odd(s(z0)), p(p(z0))), ODD(s(z0)))
COND(true, 0) → c(COND(false, p(p(p(0)))), ODD(0))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))), ODD(s(0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(p(s(s(z0)))))), ODD(s(s(z0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), p(p(0))), ODD(0))
COND(true, s(z0)) → c(COND(odd(s(z0)), p(p(z0))), ODD(s(z0)))
COND(true, 0) → c(COND(false, p(p(p(0)))), ODD(0))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))), ODD(s(0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(p(s(s(z0)))))), ODD(s(s(z0))))
K tuples:none
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c

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

Removed 1 trailing nodes:

COND(true, 0) → c(COND(false, p(p(p(0)))), ODD(0))

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), p(p(0))), ODD(0))
COND(true, s(z0)) → c(COND(odd(s(z0)), p(p(z0))), ODD(s(z0)))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))), ODD(s(0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(p(s(s(z0)))))), ODD(s(s(z0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), p(p(0))), ODD(0))
COND(true, s(z0)) → c(COND(odd(s(z0)), p(p(z0))), ODD(s(z0)))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))), ODD(s(0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(p(s(s(z0)))))), ODD(s(s(z0))))
K tuples:none
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c

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

Removed 2 trailing tuple parts

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(z0)) → c(COND(odd(s(z0)), p(p(z0))), ODD(s(z0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(p(s(s(z0)))))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(z0)) → c(COND(odd(s(z0)), p(p(z0))), ODD(s(z0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(p(s(s(z0)))))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
K tuples:none
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

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

COND(true, s(z0)) → c(COND(odd(s(z0)), p(p(z0))), ODD(s(z0)))
We considered the (Usable) Rules:

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

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

POL(0) = 0
POL(COND(x1, x2)) = [2]x2
POL(ODD(x1)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(false) = [2]
POL(odd(x1)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [2] + x1
POL(true) = 0

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(z0)) → c(COND(odd(s(z0)), p(p(z0))), ODD(s(z0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(p(s(s(z0)))))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(p(s(s(z0)))))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
K tuples:

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

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

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

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

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(p(s(s(z0)))))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)), ODD(s(0)))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))), ODD(s(0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(p(s(s(z0)))))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
K tuples:

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

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

Removed 2 trailing tuple parts

### (20) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(p(s(s(z0)))))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(p(s(s(z0)))))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
K tuples:

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

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

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

COND(true, s(s(x0))) → c(COND(odd(x0), p(p(s(x0)))), ODD(s(s(x0))))
COND(true, s(s(0))) → c(COND(false, p(p(p(s(s(0)))))), ODD(s(s(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(0))) → c(COND(false, p(p(p(s(s(0)))))), ODD(s(s(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(x0))) → c(COND(odd(x0), p(p(s(x0)))), ODD(s(s(x0))))
COND(true, s(s(0))) → c(COND(false, p(p(p(s(s(0)))))), ODD(s(s(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
K tuples:

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

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

Removed 1 trailing tuple parts

### (24) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(x0))) → c(COND(odd(x0), p(p(s(x0)))), ODD(s(s(x0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
K tuples:

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

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

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

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
We considered the (Usable) Rules:none
And the Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

### (26) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(x0))) → c(COND(odd(x0), p(p(s(x0)))), ODD(s(s(x0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(odd(s(z0)), p(p(z0))), ODD(s(z0)))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

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

COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
We considered the (Usable) Rules:

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

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND(x1, x2)) = [4] + [4]x2
POL(ODD(x1)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(false) = [1]
POL(odd(x1)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [2] + x1
POL(true) = 0

### (28) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), p(p(0))))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
K tuples:

COND(true, s(z0)) → c(COND(odd(s(z0)), p(p(z0))), ODD(s(z0)))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

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

COND(true, 0) → c(COND(odd(0), p(0)))
COND(true, 0) → c(COND(false, p(p(0))))

### (30) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, 0) → c(COND(odd(0), p(0)))
COND(true, 0) → c(COND(false, p(p(0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, 0) → c(COND(odd(0), p(0)))
COND(true, 0) → c(COND(false, p(p(0))))
K tuples:

COND(true, s(z0)) → c(COND(odd(s(z0)), p(p(z0))), ODD(s(z0)))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

Removed 1 trailing tuple parts

### (32) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, 0) → c(COND(odd(0), p(0)))
COND(true, 0) → c
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, 0) → c(COND(odd(0), p(0)))
COND(true, 0) → c
K tuples:

COND(true, s(z0)) → c(COND(odd(s(z0)), p(p(z0))), ODD(s(z0)))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c, c

### (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, 0) → c
We considered the (Usable) Rules:none
And the Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, 0) → c(COND(odd(0), p(0)))
COND(true, 0) → c
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [5]
POL(COND(x1, x2)) = [2]
POL(ODD(x1)) = 0
POL(c) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(false) = [4]
POL(odd(x1)) = [2] + x1
POL(p(x1)) = [5] + [5]x1
POL(s(x1)) = x1
POL(true) = [3]

### (34) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, 0) → c(COND(odd(0), p(0)))
COND(true, 0) → c
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(0)) → c(COND(true, p(p(p(s(0))))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, 0) → c(COND(odd(0), p(0)))
K tuples:

COND(true, s(z0)) → c(COND(odd(s(z0)), p(p(z0))), ODD(s(z0)))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, 0) → c
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c, c

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

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

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

### (36) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, 0) → c(COND(odd(0), p(0)))
COND(true, 0) → c
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, 0) → c(COND(odd(0), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
K tuples:

COND(true, s(z0)) → c(COND(odd(s(z0)), p(p(z0))), ODD(s(z0)))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, 0) → c
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c, c

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

Removed 1 trailing nodes:

COND(true, 0) → c

### (38) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, 0) → c(COND(odd(0), p(0)))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, 0) → c(COND(odd(0), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

### (39) 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)) → c(COND(true, p(p(0))))
We considered the (Usable) Rules:

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

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, 0) → c(COND(odd(0), p(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND(x1, x2)) = [4]x2
POL(ODD(x1)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(false) = [3]
POL(odd(x1)) = [1]
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = 0

### (40) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, 0) → c(COND(odd(0), p(0)))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, 0) → c(COND(odd(0), p(0)))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

Use forward instantiation to replace ODD(s(s(z0))) → c3(ODD(z0)) by

ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))

### (42) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, 0) → c(COND(odd(0), p(0)))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
S tuples:

COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, 0) → c(COND(odd(0), p(0)))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

Removed 1 trailing nodes:

COND(true, s(s(0))) → c(ODD(s(s(0))))

### (44) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, 0) → c(COND(odd(0), p(0)))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
S tuples:

COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, 0) → c(COND(odd(0), p(0)))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

Removed 1 trailing tuple parts

### (46) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, 0) → c(COND(odd(0), p(0)))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, 0) → c(COND(odd(0), p(0)))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

Used rewriting to replace COND(true, s(s(z0))) → c(COND(odd(s(s(z0))), p(z0)), ODD(s(s(z0)))) by COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))

### (48) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, 0) → c(COND(odd(0), p(0)))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, 0) → c(COND(odd(0), p(0)))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

Used rewriting to replace COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0)))) by COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))

### (50) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), p(0)))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, 0) → c(COND(odd(0), p(0)))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, 0) → c(COND(odd(0), p(0)))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

Used rewriting to replace COND(true, s(0)) → c(COND(odd(s(0)), p(0))) by COND(true, s(0)) → c(COND(odd(s(0)), 0))

### (52) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, 0) → c(COND(odd(0), p(0)))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), 0))
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, 0) → c(COND(odd(0), p(0)))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

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

COND(true, 0) → c(COND(odd(0), 0))
COND(true, 0) → c(COND(false, p(0)))

### (54) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), 0))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, 0) → c(COND(false, p(0)))
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, 0) → c(COND(false, p(0)))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

Removed 1 trailing tuple parts

### (56) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), 0))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, 0) → c
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, 0) → c
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3, c

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

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

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

COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), 0))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, 0) → c
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]
POL(COND(x1, x2)) = [4]
POL(ODD(x1)) = [2]x1
POL(c) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(false) = [3]
POL(odd(x1)) = [4]
POL(p(x1)) = [2]x1
POL(s(x1)) = 0
POL(true) = [3]

### (58) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), 0))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, 0) → c
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, 0) → c(COND(odd(0), 0))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, 0) → c
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3, c

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

Use narrowing to replace COND(true, 0) → c(COND(odd(0), 0)) by

COND(true, 0) → c(COND(false, 0))

### (60) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), 0))
COND(true, 0) → c
COND(true, 0) → c(COND(false, 0))
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, 0) → c(COND(false, 0))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, 0) → c
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3, c

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

Removed 1 trailing nodes:

COND(true, 0) → c

### (62) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(x0))) → c(COND(odd(x0), p(p(s(x0)))), ODD(s(s(x0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), 0))
COND(true, 0) → c(COND(false, 0))
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, 0) → c(COND(false, 0))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

Removed 1 trailing tuple parts

### (64) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(x0))) → c(COND(odd(x0), p(p(s(x0)))), ODD(s(s(x0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), 0))
COND(true, 0) → c
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, 0) → c
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3, c

### (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, 0) → c
We considered the (Usable) Rules:none
And the Tuples:

COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(x0))) → c(COND(odd(x0), p(p(s(x0)))), ODD(s(s(x0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), 0))
COND(true, 0) → c
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [4]
POL(COND(x1, x2)) = [4]
POL(ODD(x1)) = 0
POL(c) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(false) = [3]
POL(odd(x1)) = [2]
POL(p(x1)) = [3] + [4]x1
POL(s(x1)) = x1
POL(true) = [3]

### (66) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(x0))) → c(COND(odd(x0), p(p(s(x0)))), ODD(s(s(x0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), 0))
COND(true, 0) → c
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, 0) → c
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3, c

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

Used rewriting to replace COND(true, s(0)) → c(COND(true, p(p(0)))) by COND(true, s(0)) → c(COND(true, p(0)))

### (68) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(x0))) → c(COND(odd(x0), p(p(s(x0)))), ODD(s(s(x0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(odd(s(0)), 0))
COND(true, 0) → c
COND(true, s(0)) → c(COND(true, p(0)))
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, 0) → c
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3, c

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

Removed 2 trailing nodes:

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

### (70) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(x0))) → c(COND(odd(x0), p(p(s(x0)))), ODD(s(s(x0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(0)))
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

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

COND(true, s(0)) → c(COND(true, 0))

### (72) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(x0))) → c(COND(odd(x0), p(p(s(x0)))), ODD(s(s(x0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, 0))
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

Removed 1 trailing nodes:

COND(true, s(0)) → c(COND(true, 0))

### (74) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(x0))) → c(COND(odd(x0), p(p(s(x0)))), ODD(s(s(x0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

Used rewriting to replace COND(true, s(s(x0))) → c(COND(odd(x0), p(p(s(x0)))), ODD(s(s(x0)))) by COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))

### (76) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

Used rewriting to replace COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(p(s(s(s(s(z0)))))))), ODD(s(s(s(s(z0)))))) by COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))

### (78) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
S tuples:

ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c3, c

### (79) 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(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
We considered the (Usable) Rules:

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

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [4]
POL(COND(x1, x2)) = [2] + [4]x1 + [4]x2
POL(ODD(x1)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(false) = [2]
POL(odd(x1)) = [4]
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = [2]

### (80) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
S tuples:

ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c3, c

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

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

COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))

### (82) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(0)) → c(COND(true, p(p(0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
S tuples:

ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c3, c

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

Used rewriting to replace COND(true, s(0)) → c(COND(true, p(p(0)))) by COND(true, s(0)) → c(COND(true, p(0)))

### (84) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
COND(true, s(0)) → c(COND(true, p(0)))
S tuples:

ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

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

COND(true, s(0)) → c(COND(true, 0))

### (86) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
COND(true, s(0)) → c(COND(true, 0))
S tuples:

ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), p(p(s(z0)))), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, p(p(0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

Removed 1 trailing nodes:

COND(true, s(0)) → c(COND(true, 0))

### (88) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
S tuples:

ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
K tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

Use forward instantiation to replace ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0)))) by

ODD(s(s(s(s(s(s(y0))))))) → c3(ODD(s(s(s(s(y0))))))

### (90) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
ODD(s(s(s(s(s(s(y0))))))) → c3(ODD(s(s(s(s(y0))))))
S tuples:

COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0))))))))
ODD(s(s(s(s(s(s(y0))))))) → c3(ODD(s(s(s(s(y0))))))
K tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

Used rewriting to replace COND(true, s(s(s(0)))) → c(COND(true, p(p(p(s(s(s(0)))))))) by COND(true, s(s(s(0)))) → c(COND(true, p(p(s(s(0))))))

### (92) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
ODD(s(s(s(s(s(s(y0))))))) → c3(ODD(s(s(s(s(y0))))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(s(s(0))))))
S tuples:

ODD(s(s(s(s(s(s(y0))))))) → c3(ODD(s(s(s(s(y0))))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(s(s(0))))))
K tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

### (93) 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(0)))) → c(COND(true, p(p(s(s(0))))))
We considered the (Usable) Rules:

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

COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
ODD(s(s(s(s(s(s(y0))))))) → c3(ODD(s(s(s(s(y0))))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(s(s(0))))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(COND(x1, x2)) = [2]x2
POL(ODD(x1)) = [3]
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(false) = [5]
POL(odd(x1)) = [5]
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = [2]

### (94) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
ODD(s(s(s(s(s(s(y0))))))) → c3(ODD(s(s(s(s(y0))))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(s(s(0))))))
S tuples:

ODD(s(s(s(s(s(s(y0))))))) → c3(ODD(s(s(s(s(y0))))))
K tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(s(s(0))))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

### (95) 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.

ODD(s(s(s(s(s(s(y0))))))) → c3(ODD(s(s(s(s(y0))))))
We considered the (Usable) Rules:

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

COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
ODD(s(s(s(s(s(s(y0))))))) → c3(ODD(s(s(s(s(y0))))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(s(s(0))))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [3]
POL(COND(x1, x2)) = [3] + x22
POL(ODD(x1)) = [2] + x1
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(false) = 0
POL(odd(x1)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [1] + x1
POL(true) = 0

### (96) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), p(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
ODD(s(s(s(s(s(s(y0))))))) → c3(ODD(s(s(s(s(y0))))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(s(s(0))))))
S tuples:none
K tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(p(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(s(y0))))) → c(ODD(s(s(s(s(y0))))))
COND(true, s(s(s(0)))) → c(COND(true, p(p(s(s(0))))))
ODD(s(s(s(s(s(s(y0))))))) → c3(ODD(s(s(s(s(y0))))))
Defined Rule Symbols:

odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

The set S is empty