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

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

The problem is match-bounded by 2. The enriched problem is
compatible with the following automaton.
{ f_0(2, 2) -> 1
, f_1(2, 2) -> 3
, g_0(2, 2) -> 1
, g_1(2, 2) -> 1
, g_2(2, 2) -> 3
, h_0(2) -> 2
, h_1(1) -> 1
, h_1(1) -> 3
, h_1(3) -> 1
, h_1(3) -> 3 }

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