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

(0) Obligation:

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

not(true) → false
not(false) → true
odd(0) → false
odd(s(x)) → not(odd(x))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(s(x), y) → s(+(x, y))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(2) BOUNDS(n^1, INF)

(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
odd(0') → false
odd(s(x)) → not(odd(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(s(x), y) → s(+'(x, y))

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
not(true) → false
not(false) → true
odd(0') → false
odd(s(x)) → not(odd(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(s(x), y) → s(+'(x, y))

Types:
not :: true:false → true:false
true :: true:false
false :: true:false
odd :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':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:
odd, +'

(8) Obligation:

TRS:
Rules:
not(true) → false
not(false) → true
odd(0') → false
odd(s(x)) → not(odd(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(s(x), y) → s(+'(x, y))

Types:
not :: true:false → true:false
true :: true:false
false :: true:false
odd :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':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:
odd, +'

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

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

(10) Obligation:

TRS:
Rules:
not(true) → false
not(false) → true
odd(0') → false
odd(s(x)) → not(odd(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(s(x), y) → s(+'(x, y))

Types:
not :: true:false → true:false
true :: true:false
false :: true:false
odd :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':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:
+'

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n321_0)) → gen_0':s3_0(+(n321_0, a)), rt ∈ Ω(1 + n321₀)

Induction Base:
+'(gen_0':s3_0(a), gen_0':s3_0(0)) →_R^Ω(1)
gen_0':s3_0(a)

Induction Step:
+'(gen_0':s3_0(a), gen_0':s3_0(+(n321_0, 1))) →_R^Ω(1)
s(+'(gen_0':s3_0(a), gen_0':s3_0(n321_0))) →_IH
s(gen_0':s3_0(+(a, c322_0)))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
not(true) → false
not(false) → true
odd(0') → false
odd(s(x)) → not(odd(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(s(x), y) → s(+'(x, y))

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

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n321_0)) → gen_0':s3_0(+(n321_0, a)), rt ∈ Ω(1 + n321₀)

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.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n321_0)) → gen_0':s3_0(+(n321_0, a)), rt ∈ Ω(1 + n321₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
not(true) → false
not(false) → true
odd(0') → false
odd(s(x)) → not(odd(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(s(x), y) → s(+'(x, y))

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

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n321_0)) → gen_0':s3_0(+(n321_0, a)), rt ∈ Ω(1 + n321₀)

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.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n321_0)) → gen_0':s3_0(+(n321_0, a)), rt ∈ Ω(1 + n321₀)

(18) BOUNDS(n^1, INF)