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

Strict Trs:
  { g(s(x)) -> f(x)
  , g(0()) -> 0()
  , f(s(x)) -> s(s(g(x)))
  , f(0()) -> s(0()) }
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(g) = {}, safe(s) = {1}, safe(f) = {}, safe(0) = {}

and precedence

 g ~ f .

Following symbols are considered recursive:

 {g, f}

The recursion depth is 1.

For your convenience, here are the satisfied ordering constraints:

  g(s(; x);) > f(x;)          
                              
     g(0();) > 0()            
                              
  f(s(; x);) > s(; s(; g(x;)))
                              
     f(0();) > s(; 0())       
                              

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