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

Strict Trs:
  { f(f(x)) -> f(c(f(x)))
  , f(f(x)) -> f(d(f(x)))
  , g(c(x)) -> x
  , g(c(h(0()))) -> g(d(1()))
  , g(c(1())) -> g(d(h(0())))
  , g(d(x)) -> x
  , g(h(x)) -> g(x) }
Obligation:
  runtime complexity
Answer:
  YES(?,O(n^1))

The problem is match-bounded by 1. The enriched problem is
compatible with the following automaton.
{ f_0(2) -> 1
, c_0(2) -> 1
, c_0(2) -> 2
, d_0(2) -> 1
, d_0(2) -> 2
, d_1(4) -> 3
, g_0(2) -> 1
, g_1(2) -> 1
, g_1(3) -> 1
, h_0(2) -> 1
, h_0(2) -> 2
, h_1(5) -> 1
, h_1(5) -> 4
, 0_0() -> 1
, 0_0() -> 2
, 0_1() -> 5
, 1_0() -> 1
, 1_0() -> 2
, 1_1() -> 1
, 1_1() -> 4 }

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