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)))) → 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: 1128
Accept states: [1129, 1130]
Transitions:
1128→1129[a_1|0]
1128→1130[b_1|0]
1128→1128[s_1|0]
1128→1131[a_1|1]
1131→1129[s_1|1]
1131→1131[s_1|1]

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