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

Start state: 213
Accept states: [214, 215]
Transitions:
213→214[a_1|0]
213→215[b_1|0]
213→213[c_1|0]
213→216[a_1|1]
213→217[b_1|2]
216→215[a_1|1]
216→217[a_1|1]
217→215[b_1|2]
217→217[b_1|2]

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