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

(0) Obligation:

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

active(f(f(a))) → mark(c(f(g(f(a)))))
active(f(X)) → f(active(X))
active(g(X)) → g(active(X))
f(mark(X)) → mark(f(X))
g(mark(X)) → mark(g(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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 2.
The certificate found is represented by the following graph.

Start state: 3411
Accept states: [3412, 3413, 3414, 3415, 3416, 3417]
Transitions:
3411→3412[active_1|0]
3411→3413[f_1|0]
3411→3414[g_1|0]
3411→3415[proper_1|0]
3411→3416[c_1|0]
3411→3417[top_1|0]
3411→3411[a|0, mark_1|0, ok_1|0]
3411→3418[a|1]
3411→3419[f_1|1]
3411→3420[g_1|1]
3411→3421[proper_1|1]
3411→3422[f_1|1]
3411→3423[g_1|1]
3411→3424[c_1|1]
3411→3425[active_1|1]
3418→3415[ok_1|1]
3418→3421[ok_1|1]
3418→3426[active_1|2]
3419→3413[mark_1|1]
3419→3419[mark_1|1]
3419→3422[mark_1|1]
3420→3414[mark_1|1]
3420→3420[mark_1|1]
3420→3423[mark_1|1]
3421→3417[top_1|1]
3422→3413[ok_1|1]
3422→3419[ok_1|1]
3422→3422[ok_1|1]
3423→3414[ok_1|1]
3423→3420[ok_1|1]
3423→3423[ok_1|1]
3424→3416[ok_1|1]
3424→3424[ok_1|1]
3425→3417[top_1|1]
3426→3417[top_1|2]

(2) BOUNDS(O(1), O(n^1))