The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

a(b(x)) → b(a(x))
a(c(x)) → x

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: 1
Accept states: [2]
Transitions:
1→2[a_1|0, b_1|1, c_1|1]
1→3[b_1|1]
2→2[b_1|0, c_1|0]
3→2[a_1|1, b_1|1, c_1|1]
3→3[b_1|1]

(2) BOUNDS(1, n^1)