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:

a__f(f(a)) → a__f(g(f(a)))
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(g(X)) → g(X)
a__f(X) → f(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 3.
The certificate found is represented by the following graph.

Start state: 1
Accept states: [2]
Transitions:
1→2[a__f_1|0, mark_1|0, f_1|1, a|1, g_1|1]
1→3[a__f_1|1, f_1|2]
1→6[a__f_1|1, f_1|2]
1→7[a__f_1|2, f_1|3]
2→2[f_1|0, a|0, g_1|0]
3→4[g_1|1]
4→5[f_1|1]
5→2[a|1]
6→2[mark_1|1, a|1, g_1|1]
6→6[a__f_1|1, f_1|2]
6→7[a__f_1|2, f_1|3]
7→8[g_1|2]
8→9[f_1|2]
9→2[a|2]

(2) BOUNDS(1, n^1)