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:

a(d(x1)) → d(b(c(b(d(x1)))))
a(x1) → b(b(f(b(b(x1)))))
b(d(b(x1))) → a(d(x1))
d(f(x1)) → b(d(x1))

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: 380
Accept states: [381, 382, 383]
Transitions:
380→381[a_1|0]
380→382[b_1|0]
380→383[d_1|0]
380→380[c_1|0, f_1|0]
380→384[b_1|1]
380→388[d_1|1]
384→385[b_1|1]
385→386[f_1|1]
386→387[b_1|1]
387→381[b_1|1]
388→383[b_1|1]
388→388[b_1|1]

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