YES(?,O(n^1))

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict Trs:
  { a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
  , f(a(x), a(y)) -> a(f(x, y))
  , f(b(x), b(y)) -> b(f(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The problem is match-bounded by 1. The enriched problem is
compatible with the following automaton.
{ a_0(2) -> 1
, f_0(2, 2) -> 1
, f_1(2, 2) -> 3
, b_0(2) -> 2
, b_1(3) -> 1
, b_1(3) -> 3 }

Hurray, we answered YES(?,O(n^1))