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:

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: 1
Accept states: [2]
Transitions:
1→2[active_1|0, f_1|0, g_1|0, proper_1|0, c_1|0, top_1|0]
1→3[mark_1|1]
1→4[ok_1|1]
1→5[mark_1|1]
1→6[ok_1|1]
1→7[ok_1|1]
1→8[ok_1|1]
1→9[top_1|1]
1→10[top_1|1]
1→11[top_1|2]
2→2[a|0, mark_1|0, ok_1|0]
3→2[f_1|1]
3→3[mark_1|1]
3→4[ok_1|1]
4→2[f_1|1]
4→3[mark_1|1]
4→4[ok_1|1]
5→2[g_1|1]
5→5[mark_1|1]
5→6[ok_1|1]
6→2[g_1|1]
6→5[mark_1|1]
6→6[ok_1|1]
7→2[a|1]
8→2[c_1|1]
8→8[ok_1|1]
9→2[proper_1|1]
9→7[ok_1|1]
10→2[active_1|1]
11→7[active_1|2]

(2) BOUNDS(1, n^1)