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:
  runtime complexity
Answer:
  YES(?,O(n^1))

The input is overlay and right-linear. Switching to innermost
rewriting.

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(3, 3) -> 1
, f_1(3, 3) -> 4
, g_0(3, 3) -> 2
, g_1(3, 3) -> 1
, g_2(3, 3) -> 4
, h_0(3) -> 3
, h_1(1) -> 1
, h_1(1) -> 2
, h_1(1) -> 4
, h_1(4) -> 1
, h_1(4) -> 2
, h_1(4) -> 4 }

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