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(a(x1))) → a(b(x1))
b(c(x1)) → c(c(a(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: 121
Accept states: [122, 123]
Transitions:
121→122[a_1|0]
121→123[b_1|0]
121→121[c_1|0]
121→124[a_1|1]
121→128[b_1|2]
124→125[a_1|1]
125→126[a_1|1]
126→127[c_1|1]
127→123[c_1|1]
127→128[c_1|1]
128→126[a_1|2]

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