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

Start state: 556
Accept states: [557, 558]
Transitions:
556→557[a_1|0]
556→558[b_1|0]
556→556[c_1|0]
556→559[a_1|1]
556→560[b_1|2]
559→560[a_1|1]
560→558[b_1|1]
560→560[b_1|1]

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