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:

b(a(x1)) → a(a(d(x1)))
a(c(x1)) → b(b(x1))
d(a(b(x1))) → b(d(d(c(x1))))
d(x1) → a(x1)
b(a(c(a(x1)))) → 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: 838
Accept states: [839, 840, 841]
Transitions:
838→839[b_1|0]
838→840[a_1|0]
838→841[d_1|0, a_1|1]
838→838[c_1|0]
838→842[b_1|1]
842→840[b_1|1]
842→841[b_1|1]

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