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

(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 CpxTRS 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:

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

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

Removed 1 trailing tuple parts

(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(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

(5) CdtPolyRedPairProof (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   

(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(z0, 0) → c2(EVENODD(z0, s(0)))

K tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))

Defined Rule Symbols:

not, evenodd

Defined Pair Symbols:

EVENODD

Compound Symbols:

c4, c2

(7) 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

(8) BOUNDS(O(1), O(1))