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)