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:

f(S(x), x2) → f(x2, x)
f(0, x2) → 0

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
f(S(x), S(x1_1)) →⁺ f(x, x1_1)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / S(x), x1_1 / S(x1_1)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)