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

Start state: 328
Accept states: [329, 330]
Transitions:
328→329[a_1|0]
328→330[c_1|0]
328→328[b_1|0]
328→331[a_1|1]
328→332[a_1|2]
328→333[c_1|2]
331→332[a_1|1]
331→333[c_1|2]
332→333[c_1|1]
333→329[b_1|1]
333→331[b_1|1]
333→332[b_1|1]
333→334[a_1|2]
333→335[a_1|3]
333→336[c_1|3]
334→335[a_1|2]
334→336[c_1|3]
335→336[c_1|2]
336→332[b_1|2]

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