The Runtime Complexity (innermost) 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:

odd(S(x)) → even(x)
even(S(x)) → odd(x)
odd(0) → 0
even(0) → S(0)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
odd(S(S(x2_1))) →⁺ odd(x2_1)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x2_1 / S(S(x2_1))].
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:

odd(S(x)) → even(x)
even(S(x)) → odd(x)
odd(0') → 0'
even(0') → S(0')

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
odd(S(x)) → even(x)
even(S(x)) → odd(x)
odd(0') → 0'
even(0') → S(0')

Types:
odd :: S:0' → S:0'
S :: S:0' → S:0'
even :: S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

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

Heuristically decided to analyse the following defined symbols:
odd, even

They will be analysed ascendingly in the following order:
odd = even

(8) Obligation:

Innermost TRS:
Rules:
odd(S(x)) → even(x)
even(S(x)) → odd(x)
odd(0') → 0'
even(0') → S(0')

Types:
odd :: S:0' → S:0'
S :: S:0' → S:0'
even :: S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

The following defined symbols remain to be analysed:
even, odd

They will be analysed ascendingly in the following order:
odd = even

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
even(gen_S:0'2_0(*(2, n4_0))) → gen_S:0'2_0(1), rt ∈ Ω(1 + n4₀)

Induction Base:
even(gen_S:0'2_0(*(2, 0))) →_R^Ω(1)
S(0')

Induction Step:
even(gen_S:0'2_0(*(2, +(n4_0, 1)))) →_R^Ω(1)
odd(gen_S:0'2_0(+(1, *(2, n4_0)))) →_R^Ω(1)
even(gen_S:0'2_0(*(2, n4_0))) →_IH
gen_S:0'2_0(1)

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
odd(S(x)) → even(x)
even(S(x)) → odd(x)
odd(0') → 0'
even(0') → S(0')

Types:
odd :: S:0' → S:0'
S :: S:0' → S:0'
even :: S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Lemmas:
even(gen_S:0'2_0(*(2, n4_0))) → gen_S:0'2_0(1), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

The following defined symbols remain to be analysed:
odd

They will be analysed ascendingly in the following order:
odd = even

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

Innermost TRS:
Rules:
odd(S(x)) → even(x)
even(S(x)) → odd(x)
odd(0') → 0'
even(0') → S(0')

Types:
odd :: S:0' → S:0'
S :: S:0' → S:0'
even :: S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Lemmas:
even(gen_S:0'2_0(*(2, n4_0))) → gen_S:0'2_0(1), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
even(gen_S:0'2_0(*(2, n4_0))) → gen_S:0'2_0(1), rt ∈ Ω(1 + n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost TRS:
Rules:
odd(S(x)) → even(x)
even(S(x)) → odd(x)
odd(0') → 0'
even(0') → S(0')

Types:
odd :: S:0' → S:0'
S :: S:0' → S:0'
even :: S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Lemmas:
even(gen_S:0'2_0(*(2, n4_0))) → gen_S:0'2_0(1), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
even(gen_S:0'2_0(*(2, n4_0))) → gen_S:0'2_0(1), rt ∈ Ω(1 + n4₀)

(18) BOUNDS(n^1, INF)