YES(?,O(1))

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

Strict Trs:
  { f(X) -> n__f(X)
  , f(f(a())) -> f(g(n__f(a())))
  , activate(X) -> X
  , activate(n__f(X)) -> f(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

Arguments of following rules are not normal-forms:

{ f(f(a())) -> f(g(n__f(a()))) }

All above mentioned rules can be savely removed.

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

Strict Trs:
  { f(X) -> n__f(X)
  , activate(X) -> X
  , activate(n__f(X)) -> 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(f) = {}, safe(a) = {}, safe(g) = {1}, safe(n__f) = {1},
 safe(activate) = {}

and precedence

 activate > f .

Following symbols are considered recursive:

 {}

The recursion depth is 0.

For your convenience, here are the satisfied ordering constraints:

                 f(X;) > n__f(; X)
                                  
          activate(X;) > X        
                                  
  activate(n__f(; X);) > f(X;)    
                                  

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