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:

f(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → 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[f_1|0, h_1|0, activate_1|0, n__f_1|1, n__h_1|1, g_1|1]
1→3[g_1|1]
1→5[h_1|1, n__h_1|2]
1→6[f_1|1, n__f_1|2]
1→7[g_1|2]
2→2[g_1|0, n__h_1|0, n__f_1|0]
3→4[n__h_1|1]
4→2[n__f_1|1]
5→2[activate_1|1, n__h_1|1, n__f_1|1, g_1|1]
5→5[h_1|1, n__h_1|2]
5→6[f_1|1, n__f_1|2]
5→7[g_1|2]
6→2[activate_1|1, n__h_1|1, n__f_1|1, g_1|1]
6→5[h_1|1, n__h_1|2]
6→6[f_1|1, n__f_1|2]
6→7[g_1|2]
7→8[n__h_1|2]
8→6[n__f_1|2]

(2) BOUNDS(1, n^1)