### (0) Obligation:

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

not(true) → false
not(false) → true
evenodd(x, 0) → not(evenodd(x, s(0)))
evenodd(0, s(0)) → false
evenodd(s(x), s(0)) → evenodd(x, 0)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

not(true) → false
not(false) → true
evenodd(z0, 0) → not(evenodd(z0, s(0)))
evenodd(0, s(0)) → false
evenodd(s(z0), s(0)) → evenodd(z0, 0)
Tuples:

NOT(true) → c
NOT(false) → c1
EVENODD(z0, 0) → c2(NOT(evenodd(z0, s(0))), EVENODD(z0, s(0)))
EVENODD(0, s(0)) → c3
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
S tuples:

NOT(true) → c
NOT(false) → c1
EVENODD(z0, 0) → c2(NOT(evenodd(z0, s(0))), EVENODD(z0, s(0)))
EVENODD(0, s(0)) → c3
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
K tuples:none
Defined Rule Symbols:

not, evenodd

Defined Pair Symbols:

NOT, EVENODD

Compound Symbols:

c, c1, c2, c3, c4

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

Removed 3 trailing nodes:

NOT(true) → c
NOT(false) → c1
EVENODD(0, s(0)) → c3

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

not(true) → false
not(false) → true
evenodd(z0, 0) → not(evenodd(z0, s(0)))
evenodd(0, s(0)) → false
evenodd(s(z0), s(0)) → evenodd(z0, 0)
Tuples:

EVENODD(z0, 0) → c2(NOT(evenodd(z0, s(0))), EVENODD(z0, s(0)))
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
S tuples:

EVENODD(z0, 0) → c2(NOT(evenodd(z0, s(0))), EVENODD(z0, s(0)))
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
K tuples:none
Defined Rule Symbols:

not, evenodd

Defined Pair Symbols:

EVENODD

Compound Symbols:

c2, c4

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

Removed 1 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

not(true) → false
not(false) → true
evenodd(z0, 0) → not(evenodd(z0, s(0)))
evenodd(0, s(0)) → false
evenodd(s(z0), s(0)) → evenodd(z0, 0)
Tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
S tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
K tuples:none
Defined Rule Symbols:

not, evenodd

Defined Pair Symbols:

EVENODD

Compound Symbols:

c4, c2

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

not(true) → false
not(false) → true
evenodd(z0, 0) → not(evenodd(z0, s(0)))
evenodd(0, s(0)) → false
evenodd(s(z0), s(0)) → evenodd(z0, 0)

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
S tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

EVENODD

Compound Symbols:

c4, c2

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

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

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
We considered the (Usable) Rules:none
And the Tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(EVENODD(x1, x2)) = x1
POL(c2(x1)) = x1
POL(c4(x1)) = x1
POL(s(x1)) = [1] + x1

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
S tuples:

EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
K tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
Defined Rule Symbols:none

Defined Pair Symbols:

EVENODD

Compound Symbols:

c4, c2

### (11) CdtKnowledgeProof (EQUIVALENT transformation)

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

EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
Now S is empty