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: 5087
Accept states: [5088]
Transitions:
5087→5088[active_1|0, f_1|0, h_1|0, proper_1|0, c_1|0, g_1|0, d_1|0, top_1|0]
5087→5089[mark_1|1]
5087→5090[ok_1|1]
5087→5091[mark_1|1]
5087→5092[ok_1|1]
5087→5093[ok_1|1]
5087→5094[ok_1|1]
5087→5095[ok_1|1]
5087→5096[top_1|1]
5087→5097[top_1|1]
5088→5088[mark_1|0, ok_1|0]
5089→5088[f_1|1]
5089→5089[mark_1|1]
5089→5090[ok_1|1]
5090→5088[f_1|1]
5090→5089[mark_1|1]
5090→5090[ok_1|1]
5091→5088[h_1|1]
5091→5091[mark_1|1]
5091→5092[ok_1|1]
5092→5088[h_1|1]
5092→5091[mark_1|1]
5092→5092[ok_1|1]
5093→5088[c_1|1]
5093→5093[ok_1|1]
5094→5088[g_1|1]
5094→5094[ok_1|1]
5095→5088[d_1|1]
5095→5095[ok_1|1]
5096→5088[proper_1|1]
5097→5088[active_1|1]

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