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(f(a)) → c(n__f(n__g(n__f(n__a))))
f(X) → n__f(X)
g(X) → n__g(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(n__a) → a
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, g_1|0, a|0, activate_1|0, n__f_1|1, n__g_1|1, n__a|1, a|1, c_1|1, n__a|2]
1→3[f_1|1, n__f_1|2]
1→4[g_1|1, n__g_1|2]
1→5[c_1|2]
2→2[c_1|0, n__f_1|0, n__g_1|0, n__a|0]
3→2[activate_1|1, n__f_1|1, n__g_1|1, a|1, n__a|1, c_1|1, n__a|2]
3→3[f_1|1, n__f_1|2]
3→4[g_1|1, n__g_1|2]
3→5[c_1|2]
4→2[activate_1|1, n__f_1|1, n__g_1|1, a|1, n__a|1, c_1|1, n__a|2]
4→3[f_1|1, n__f_1|2]
4→4[g_1|1, n__g_1|2]
4→5[c_1|2]
5→6[n__f_1|2]
6→7[n__g_1|2]
7→8[n__f_1|2]
8→2[n__a|2]

(2) BOUNDS(1, n^1)