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

Strict Trs:
  { f(g(x)) -> g(g(f(x)))
  , f(g(x)) -> g(g(g(x))) }
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) = {}, safe(g) = {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);) > g(; g(; f(x;))) 
                               
  f(g(; x);) > g(; g(; g(; x)))
                               

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