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(a(x1)) → a(b(c(c(x1))))
c(a(x1)) → x1
c(b(x1)) → a(a(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 3.
The certificate found is represented by the following graph.

Start state: 195
Accept states: [196, 197]
Transitions:
195→196[a_1|0]
195→197[c_1|0]
195→195[b_1|0]
195→198[a_1|1]
195→199[c_1|2]
198→197[a_1|1]
198→199[a_1|1]
198→201[b_1|2]
199→200[c_1|2]
200→201[b_1|2]
201→197[a_1|2]
201→199[a_1|2]
201→201[b_1|3]

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