The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0

Rewrite Strategy: INNERMOST

(1) CpxTrsMatchBoundsProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1.
The certificate found is represented by the following graph.

Start state: 1
Accept states: [2]
Transitions:
1→2[g_1|0, f_1|0, f_1|1, 0|1]
1→3[s_1|1]
1→4[s_1|1]
2→2[s_1|0, 0|0]
3→2[0|1]
4→5[s_1|1]
5→2[g_1|1, f_1|1, 0|1]
5→3[s_1|1]
5→4[s_1|1]

(2) BOUNDS(1, n^1)