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(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: 368
Accept states: [369, 370]
Transitions:
368→369[a_1|0, b_1|1]
368→370[b_1|0]
368→368[c_1|0]
368→371[b_1|1]
368→373[a_1|2, b_1|2, b_1|3]
368→374[c_1|2]
371→372[a_1|1, b_1|2]
371→373[a_1|2, b_1|2, b_1|3]
371→374[c_1|2]
372→373[a_1|1, b_1|2]
372→374[c_1|2]
373→374[c_1|1]
374→369[c_1|1]
374→373[c_1|1]

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