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(b(x1)) → b(r(x1))
r(a(x1)) → d(r(x1))
r(x1) → d(x1)
d(a(x1)) → a(a(d(x1)))
d(x1) → 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 3.
The certificate found is represented by the following graph.

Start state: 1248
Accept states: [1249, 1250, 1251]
Transitions:
1248→1249[a_1|0]
1248→1250[r_1|0, d_1|1, a_1|2]
1248→1251[d_1|0, a_1|1]
1248→1248[b_1|0]
1248→1252[r_1|1, d_1|2, a_1|3]
1252→1249[b_1|1]
1252→1251[b_1|1]
1252→1250[b_1|1]
1252→1252[b_1|1]

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