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

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

The input was oriented with the instance of 'Small Polynomial Path
Order (PS,1-bounded)' as induced by the safe mapping

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

and precedence

 mark > a__f .

Following symbols are considered recursive:

 {mark}

The recursion depth is 1.

For your convenience, here are the satisfied ordering constraints:

      a__f(; X) > g(; h(; f(; X)))
                                  
      a__f(; X) > f(; X)          
                                  
  mark(g(; X);) > g(; X)          
                                  
  mark(h(; X);) > h(; mark(X;))   
                                  
  mark(f(; X);) > a__f(; mark(X;))
                                  

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