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
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [true#] = 0,
    
    [activate#](x0) = 2x0 + 2,
    
    [if#](x0, x1, x2) = x0 + 2x2 + 1,
    
    [f#](x0) = x0 + 5/2,
    
    [activate](x0) = x0,
    
    [false] = 3,
    
    [true] = 0,
    
    [if](x0, x1, x2) = 2x1 + x2,
    
    [n__f](x0) = x0 + 1/2,
    
    [n__true] = 0,
    
    [c] = 0,
    
    [f](x0) = x0 + 1/2
   orientation:
    f#(X) = X + 5/2 >= X + 2 = if#(X,c(),n__f(n__true()))
    
    if#(false(),X,Y) = 2Y + 4 >= 2Y + 2 = activate#(Y)
    
    activate#(n__f(X)) = 2X + 3 >= 2X + 2 = activate#(X)
    
    activate#(n__f(X)) = 2X + 3 >= X + 5/2 = f#(activate(X))
    
    activate#(n__true()) = 2 >= 0 = true#()
    
    f(X) = X + 1/2 >= 1/2 = if(X,c(),n__f(n__true()))
    
    if(true(),X,Y) = 2X + Y >= X = X
    
    if(false(),X,Y) = 2X + Y >= Y = activate(Y)
    
    f(X) = X + 1/2 >= X + 1/2 = n__f(X)
    
    true() = 0 >= 0 = n__true()
    
    activate(n__f(X)) = X + 1/2 >= X + 1/2 = f(activate(X))
    
    activate(n__true()) = 0 >= 0 = true()
    
    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