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) → b(x1)
a(c(x1)) → c(c(a(a(b(x1)))))
b(b(x1)) → 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: 252
Accept states: [253, 254]
Transitions:
252→253[a_1|0, b_1|1]
252→254[b_1|0]
252→252[c_1|0]
252→255[b_1|1]
252→257[a_1|2, b_1|2, b_1|3]
255→256[a_1|1, b_1|2]
255→258[c_1|2]
256→257[a_1|1, b_1|2]
257→258[c_1|1]
258→253[c_1|1]
258→257[c_1|1]

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