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))) → b(x1)
b(c(x1)) → c(c(a(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: 593
Accept states: [594, 595]
Transitions:
593→594[a_1|0]
593→595[b_1|0]
593→593[c_1|0]
593→596[a_1|1]
593→598[b_1|2]
596→597[a_1|1]
596→599[b_1|2]
597→598[a_1|1]
598→599[a_1|1]
599→600[c_1|1]
600→595[c_1|1]
600→598[c_1|1]

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