YES

Problem:
 f(X) -> if(X,c(),n__f(n__true()))
 if(true(),X,Y) -> X
 if(false(),X,Y) -> activate(Y)
 f(X) -> n__f(X)
 true() -> n__true()
 activate(n__f(X)) -> f(activate(X))
 activate(n__true()) -> true()
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   f#(X) -> if#(X,c(),n__f(n__true()))
   if#(false(),X,Y) -> activate#(Y)
   activate#(n__f(X)) -> activate#(X)
   activate#(n__f(X)) -> f#(activate(X))
   activate#(n__true()) -> true#()
  TRS:
   f(X) -> if(X,c(),n__f(n__true()))
   if(true(),X,Y) -> X
   if(false(),X,Y) -> activate(Y)
   f(X) -> n__f(X)
   true() -> n__true()
   activate(n__f(X)) -> f(activate(X))
   activate(n__true()) -> true()
   activate(X) -> X
  TDG Processor:
   DPs:
    f#(X) -> if#(X,c(),n__f(n__true()))
    if#(false(),X,Y) -> activate#(Y)
    activate#(n__f(X)) -> activate#(X)
    activate#(n__f(X)) -> f#(activate(X))
    activate#(n__true()) -> true#()
   TRS:
    f(X) -> if(X,c(),n__f(n__true()))
    if(true(),X,Y) -> X
    if(false(),X,Y) -> activate(Y)
    f(X) -> n__f(X)
    true() -> n__true()
    activate(n__f(X)) -> f(activate(X))
    activate(n__true()) -> true()
    activate(X) -> X
   graph:
    activate#(n__f(X)) -> activate#(X) ->
    activate#(n__true()) -> true#()
    activate#(n__f(X)) -> activate#(X) ->
    activate#(n__f(X)) -> f#(activate(X))
    activate#(n__f(X)) -> activate#(X) ->
    activate#(n__f(X)) -> activate#(X)
    activate#(n__f(X)) -> f#(activate(X)) ->
    f#(X) -> if#(X,c(),n__f(n__true()))
    if#(false(),X,Y) -> activate#(Y) ->
    activate#(n__true()) -> true#()
    if#(false(),X,Y) -> activate#(Y) ->
    activate#(n__f(X)) -> f#(activate(X))
    if#(false(),X,Y) -> activate#(Y) ->
    activate#(n__f(X)) -> activate#(X)
    f#(X) -> if#(X,c(),n__f(n__true())) -> if#(false(),X,Y) -> activate#(Y)
   CDG Processor:
    DPs:
     f#(X) -> if#(X,c(),n__f(n__true()))
     if#(false(),X,Y) -> activate#(Y)
     activate#(n__f(X)) -> activate#(X)
     activate#(n__f(X)) -> f#(activate(X))
     activate#(n__true()) -> true#()
    TRS:
     f(X) -> if(X,c(),n__f(n__true()))
     if(true(),X,Y) -> X
     if(false(),X,Y) -> activate(Y)
     f(X) -> n__f(X)
     true() -> n__true()
     activate(n__f(X)) -> f(activate(X))
     activate(n__true()) -> true()
     activate(X) -> X
    graph:
     activate#(n__f(X)) -> activate#(X) ->
     activate#(n__f(X)) -> activate#(X)
     activate#(n__f(X)) -> activate#(X) ->
     activate#(n__f(X)) -> f#(activate(X))
     activate#(n__f(X)) -> activate#(X) ->
     activate#(n__true()) -> true#()
     activate#(n__f(X)) -> f#(activate(X)) ->
     f#(X) -> if#(X,c(),n__f(n__true()))
     if#(false(),X,Y) -> activate#(Y) ->
     activate#(n__f(X)) -> activate#(X)
     if#(false(),X,Y) -> activate#(Y) ->
     activate#(n__f(X)) -> f#(activate(X))
     if#(false(),X,Y) -> activate#(Y) -> activate#(n__true()) -> true#()
    SCC Processor:
     #sccs: 1
     #rules: 1
     #arcs: 7/25
     DPs:
      activate#(n__f(X)) -> activate#(X)
     TRS:
      f(X) -> if(X,c(),n__f(n__true()))
      if(true(),X,Y) -> X
      if(false(),X,Y) -> activate(Y)
      f(X) -> n__f(X)
      true() -> n__true()
      activate(n__f(X)) -> f(activate(X))
      activate(n__true()) -> true()
      activate(X) -> X
     Matrix Interpretation Processor: dim=3
      
      interpretation:
       [activate#](x0) = [0 1 1]x0 + [1],
       
                          
       [activate](x0) = x0
                          ,
       
                 [1]
       [false] = [0]
                 [1],
       
                [0]
       [true] = [0]
                [0],
       
                          [0 0 0]            
       [if](x0, x1, x2) = [0 0 0]x0 + x1 + x2
                          [1 0 1]            ,
       
                    [0 0 0]     [0]
       [n__f](x0) = [0 0 0]x0 + [1]
                    [1 1 1]     [0],
       
                   [0]
       [n__true] = [0]
                   [0],
       
             [0]
       [c] = [0]
             [0],
       
                 [0 0 0]     [0]
       [f](x0) = [0 0 0]x0 + [1]
                 [1 1 1]     [0]
      orientation:
       activate#(n__f(X)) = [1 1 1]X + [2] >= [0 1 1]X + [1] = activate#(X)
       
              [0 0 0]    [0]    [0 0 0]    [0]                            
       f(X) = [0 0 0]X + [1] >= [0 0 0]X + [1] = if(X,c(),n__f(n__true()))
              [1 1 1]    [0]    [1 0 1]    [0]                            
       
                                      
       if(true(),X,Y) = X + Y >= X = X
                                      
       
                                 [0]                   
       if(false(),X,Y) = X + Y + [0] >= Y = activate(Y)
                                 [2]                   
       
              [0 0 0]    [0]    [0 0 0]    [0]          
       f(X) = [0 0 0]X + [1] >= [0 0 0]X + [1] = n__f(X)
              [1 1 1]    [0]    [1 1 1]    [0]          
       
                [0]    [0]            
       true() = [0] >= [0] = n__true()
                [0]    [0]            
       
                           [0 0 0]    [0]    [0 0 0]    [0]                 
       activate(n__f(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = f(activate(X))
                           [1 1 1]    [0]    [1 1 1]    [0]                 
       
                             [0]    [0]         
       activate(n__true()) = [0] >= [0] = true()
                             [0]    [0]         
       
                               
       activate(X) = X >= X = X
                               
      problem:
       DPs:
        
       TRS:
        f(X) -> if(X,c(),n__f(n__true()))
        if(true(),X,Y) -> X
        if(false(),X,Y) -> activate(Y)
        f(X) -> n__f(X)
        true() -> n__true()
        activate(n__f(X)) -> f(activate(X))
        activate(n__true()) -> true()
        activate(X) -> X
      Qed