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

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

The input was oriented with the instance of 'Small Polynomial Path
Order (PS)' as induced by the safe mapping

 safe(f) = {1}, safe(g) = {1}, safe(h) = {1}

and precedence

 empty .

Following symbols are considered recursive:

 {f}

The recursion depth is 1.

For your convenience, here are the satisfied ordering constraints:

  f(g(; x); x) > x           
                             
  f(h(; y); x) > f(y; h(; x))
                             

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