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:

g(S(x), y) → g(x, S(y))
f(y, S(x)) → f(S(y), x)
g(0, x2) → x2
f(x1, 0) → g(x1, 0)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(2) BOUNDS(n^1, INF)