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

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

The input was oriented with the instance of 'Small Polynomial Path
Order (PS,0-bounded)' as induced by the safe mapping

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

and precedence

 a__c > a__f, mark > a__c, mark > a__f .

Following symbols are considered recursive:

 {}

The recursion depth is 0.

For your convenience, here are the satisfied ordering constraints:

          a__c() > a__f(; g(; c()))
                                   
          a__c() > c()             
                                   
       a__f(; X) > f(; X)          
                                   
  a__f(; g(; X)) > g(; X)          
                                   
  mark(; g(; X)) > g(; X)          
                                   
     mark(; c()) > a__c()          
                                   
  mark(; f(; X)) > a__f(; X)       
                                   

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