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

We add the following weak dependency pairs:

Strict DPs:
  { f^#(X) -> c_1()
  , activate^#(X) -> c_2()
  , activate^#(n__f(X)) -> c_3(f^#(X)) }

and mark the set of starting terms.

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

Strict DPs:
  { f^#(X) -> c_1()
  , activate^#(X) -> c_2()
  , activate^#(n__f(X)) -> c_3(f^#(X)) }
Strict Trs:
  { f(X) -> n__f(X)
  , activate(X) -> X
  , activate(n__f(X)) -> f(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

No rule is usable, rules are removed from the input problem.

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

Strict DPs:
  { f^#(X) -> c_1()
  , activate^#(X) -> c_2()
  , activate^#(n__f(X)) -> c_3(f^#(X)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(c_3) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

        [n__f](x1) = [1]           
                     [1]           
                                   
         [f^#](x1) = [2]           
                     [0]           
                                   
             [c_1] = [0]           
                     [0]           
                                   
  [activate^#](x1) = [1 1] x1 + [0]
                     [1 1]      [1]
                                   
             [c_2] = [0]           
                     [0]           
                                   
         [c_3](x1) = [1 0] x1 + [2]
                     [0 1]      [0]

The order satisfies the following ordering constraints:

               [f^#(X)] =  [2]          
                           [0]          
                        >  [0]          
                           [0]          
                        =  [c_1()]      
                                        
        [activate^#(X)] =  [1 1] X + [0]
                           [1 1]     [1]
                        >= [0]          
                           [0]          
                        =  [c_2()]      
                                        
  [activate^#(n__f(X))] =  [2]          
                           [3]          
                        ?  [4]          
                           [0]          
                        =  [c_3(f^#(X))]
                                        

Further, it can be verified that all rules not oriented are covered by the weightgap condition.

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

Strict DPs:
  { activate^#(X) -> c_2()
  , activate^#(n__f(X)) -> c_3(f^#(X)) }
Weak DPs: { f^#(X) -> c_1() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We estimate the number of application of {1,2} by applications of
Pre({1,2}) = {}. Here rules are labeled as follows:

  DPs:
    { 1: activate^#(X) -> c_2()
    , 2: activate^#(n__f(X)) -> c_3(f^#(X))
    , 3: f^#(X) -> c_1() }

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

Weak DPs:
  { f^#(X) -> c_1()
  , activate^#(X) -> c_2()
  , activate^#(n__f(X)) -> c_3(f^#(X)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ f^#(X) -> c_1()
, activate^#(X) -> c_2()
, activate^#(n__f(X)) -> c_3(f^#(X)) }

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

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

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