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:

f(g(x)) → g(g(f(x)))
f(g(x)) → g(g(g(x)))

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[f_1|0]
1→3[g_1|1]
1→5[g_1|1]
2→2[g_1|0]
3→4[g_1|1]
4→2[f_1|1]
4→3[g_1|1]
4→5[g_1|1]
5→6[g_1|1]
6→2[g_1|1]

(2) BOUNDS(1, n^1)