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(s(x1)) → s(a(x1))
b(a(b(s(x1)))) → a(b(s(a(x1))))
b(a(b(b(x1)))) → c(s(x1))
c(s(x1)) → a(b(a(b(x1))))
a(b(a(a(x1)))) → b(a(b(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: 1272
Accept states: [1273, 1274, 1275]
Transitions:
1272→1273[a_1|0]
1272→1274[b_1|0]
1272→1275[c_1|0]
1272→1272[s_1|0]
1272→1276[a_1|1]
1272→1277[b_1|1]
1272→1280[a_1|1]
1276→1273[s_1|1]
1276→1276[s_1|1]
1276→1280[s_1|1]
1277→1278[a_1|1]
1278→1279[b_1|1]
1279→1275[a_1|1]
1280→1281[s_1|1]
1281→1282[b_1|1]
1282→1279[a_1|1]

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