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(b(x1))
b(b(b(c(x1)))) → c(c(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: 136
Accept states: [137, 138]
Transitions:
136→137[a_1|0]
136→138[b_1|0]
136→136[c_1|0]
136→139[b_1|1]
139→137[b_1|1]

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