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(b(x1))) → a(b(c(a(x1))))
a(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 1.
The certificate found is represented by the following graph.

Start state: 179
Accept states: [180]
Transitions:
179→180[a_1|0]
179→179[b_1|0, c_1|0]
179→181[a_1|1]
179→183[a_1|1]
181→182[a_1|1]
182→180[b_1|1]
182→181[b_1|1]
182→183[b_1|1]
183→184[c_1|1]
184→185[b_1|1]
185→182[a_1|1]

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