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:

d(a(x1)) → b(d(x1))
b(x1) → a(a(a(x1)))
c(d(c(x1))) → a(d(x1))
b(d(d(x1))) → c(c(d(d(c(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 2.
The certificate found is represented by the following graph.

Start state: 857
Accept states: [858, 859, 860]
Transitions:
857→858[d_1|0]
857→859[b_1|0]
857→860[c_1|0]
857→857[a_1|0]
857→861[a_1|1]
857→863[d_1|1]
861→862[a_1|1]
862→859[a_1|1]
863→858[b_1|1]
863→863[b_1|1]
863→864[a_1|2]
864→865[a_1|2]
865→858[a_1|2]
865→863[a_1|2]

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