### (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: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
evenodd(s(x10_1), 0) →+ not(evenodd(x10_1, 0))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x10_1 / s(x10_1)].
The result substitution is [ ].

### (3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

### (4) Obligation:

Runtime Complexity Relative 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')

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
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')

Types:
not :: true:false → true:false
true :: true:false
false :: true:false
evenodd :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

### (7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
evenodd

### (8) Obligation:

TRS:
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')

Types:
not :: true:false → true:false
true :: true:false
false :: true:false
evenodd :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
evenodd

### (9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol evenodd.

### (10) Obligation:

TRS:
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')

Types:
not :: true:false → true:false
true :: true:false
false :: true:false
evenodd :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.