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:

g(x, y) → x
g(x, y) → y
f(0, 1, x) → f(s(x), x, x)
f(x, y, s(z)) → s(f(0, 1, z))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
f(0, 1, s(z16_2)) →⁺ s(f(0, 1, z16_2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [z16_2 / s(z16_2)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)