YES

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

Proof:
 DP Processor:
  DPs:
   f#(X) -> if#(X,c(),n__f(true()))
   if#(false(),X,Y) -> activate#(Y)
   activate#(n__f(X)) -> f#(X)
  TRS:
   f(X) -> if(X,c(),n__f(true()))
   if(true(),X,Y) -> X
   if(false(),X,Y) -> activate(Y)
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(X)
   activate(X) -> X
  Arctic Interpretation Processor:
   dimension: 1
   interpretation:
    [activate#](x0) = 2x0 + 6,
    
    [if#](x0, x1, x2) = x0 + 4x1 + 3x2 + 0,
    
    [f#](x0) = 1x0 + 5,
    
    [activate](x0) = 2x0 + 6,
    
    [false] = 7,
    
    [if](x0, x1, x2) = x0 + 2x1 + 3x2 + 0,
    
    [n__f](x0) = x0 + 1,
    
    [true] = 0,
    
    [c] = 0,
    
    [f](x0) = 1x0 + 5
   orientation:
    f#(X) = 1X + 5 >= X + 4 = if#(X,c(),n__f(true()))
    
    if#(false(),X,Y) = 4X + 3Y + 7 >= 2Y + 6 = activate#(Y)
    
    activate#(n__f(X)) = 2X + 6 >= 1X + 5 = f#(X)
    
    f(X) = 1X + 5 >= X + 4 = if(X,c(),n__f(true()))
    
    if(true(),X,Y) = 2X + 3Y + 0 >= X = X
    
    if(false(),X,Y) = 2X + 3Y + 7 >= 2Y + 6 = activate(Y)
    
    f(X) = 1X + 5 >= X + 1 = n__f(X)
    
    activate(n__f(X)) = 2X + 6 >= 1X + 5 = f(X)
    
    activate(X) = 2X + 6 >= X = X
   problem:
    DPs:
     
    TRS:
     f(X) -> if(X,c(),n__f(true()))
     if(true(),X,Y) -> X
     if(false(),X,Y) -> activate(Y)
     f(X) -> n__f(X)
     activate(n__f(X)) -> f(X)
     activate(X) -> X
   Qed