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)) → x1
b(b(x1)) → c(c(c(x1)))
b(c(x1)) → a(b(b(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: 376
Accept states: [377, 378]
Transitions:
376→377[a_1|0]
376→378[b_1|0]
376→376[c_1|0]
376→379[b_1|1]
376→381[c_1|2]
379→380[b_1|1]
380→378[a_1|1]
380→379[a_1|1]
381→382[c_1|2]
382→380[c_1|2]

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