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(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(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 1.
The certificate found is represented by the following graph.

Start state: 5172
Accept states: [5173, 5174, 5175, 5176, 5177, 5178, 5179, 5180]
Transitions:
5172→5173[active_1|0]
5172→5174[f_1|0]
5172→5175[h_1|0]
5172→5176[proper_1|0]
5172→5177[c_1|0]
5172→5178[g_1|0]
5172→5179[d_1|0]
5172→5180[top_1|0]
5172→5172[mark_1|0, ok_1|0]
5172→5181[f_1|1]
5172→5182[h_1|1]
5172→5183[proper_1|1]
5172→5184[f_1|1]
5172→5185[h_1|1]
5172→5186[c_1|1]
5172→5187[g_1|1]
5172→5188[d_1|1]
5172→5189[active_1|1]
5181→5174[mark_1|1]
5181→5181[mark_1|1]
5181→5184[mark_1|1]
5182→5175[mark_1|1]
5182→5182[mark_1|1]
5182→5185[mark_1|1]
5183→5180[top_1|1]
5184→5174[ok_1|1]
5184→5181[ok_1|1]
5184→5184[ok_1|1]
5185→5175[ok_1|1]
5185→5182[ok_1|1]
5185→5185[ok_1|1]
5186→5177[ok_1|1]
5186→5186[ok_1|1]
5187→5178[ok_1|1]
5187→5187[ok_1|1]
5188→5179[ok_1|1]
5188→5188[ok_1|1]
5189→5180[top_1|1]

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