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

(0) Obligation:

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

f(x, a(b(y))) → f(a(a(b(x))), y)
f(a(x), y) → f(x, a(y))
f(b(x), y) → f(x, b(y))

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
f(x, a(b(y))) →⁺ f(a(a(b(x))), y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [y / a(b(y))].
The result substitution is [x / a(a(b(x)))].

(2) BOUNDS(n^1, INF)