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(a) → f(c(a))
f(c(X)) → X
f(c(a)) → f(d(b))
f(a) → f(d(a))
f(d(X)) → X
f(c(b)) → f(d(a))
e(g(X)) → e(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 3.
The certificate found is represented by the following graph.

Start state: 1
Accept states: [2]
Transitions:
1→2[f_1|0, e_1|0, a|1, c_1|1, d_1|1, b|1, g_1|1, e_1|1, a|2, b|2, b|3]
1→3[f_1|1]
1→5[f_1|1]
1→7[f_1|1]
1→9[f_1|2]
2→2[a|0, c_1|0, d_1|0, b|0, g_1|0]
3→4[c_1|1]
4→2[a|1]
5→6[d_1|1]
6→2[a|1]
7→8[d_1|1]
8→2[b|1]
9→10[d_1|2]
10→2[b|2]

(2) BOUNDS(1, n^1)