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: 3160
Accept states: [3161]
Transitions:
3160→3161[active_1|0, f_1|0, g_1|0, proper_1|0, c_1|0, top_1|0]
3160→3162[mark_1|1]
3160→3163[ok_1|1]
3160→3164[mark_1|1]
3160→3165[ok_1|1]
3160→3166[ok_1|1]
3160→3167[ok_1|1]
3160→3168[top_1|1]
3160→3169[top_1|1]
3160→3170[top_1|2]
3161→3161[a|0, mark_1|0, ok_1|0]
3162→3161[f_1|1]
3162→3162[mark_1|1]
3162→3163[ok_1|1]
3163→3161[f_1|1]
3163→3162[mark_1|1]
3163→3163[ok_1|1]
3164→3161[g_1|1]
3164→3164[mark_1|1]
3164→3165[ok_1|1]
3165→3161[g_1|1]
3165→3164[mark_1|1]
3165→3165[ok_1|1]
3166→3161[a|1]
3167→3161[c_1|1]
3167→3167[ok_1|1]
3168→3161[proper_1|1]
3168→3166[ok_1|1]
3169→3161[active_1|1]
3170→3166[active_1|2]

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