YES
TRS:
 {   if(true(), X, Y) -> X,
    if(false(), X, Y) -> activate(Y),
                 f(X) -> if(X, c(), n__f(n__true())),
                 f(X) -> n__f(X),
               true() -> n__true(),
          activate(X) -> X,
    activate(n__f(X)) -> f(activate(X)),
  activate(n__true()) -> true()}
 RUF:
  Strict:
   {   if(true(), X, Y) -> X,
                   f(X) -> if(X, c(), n__f(n__true())),
                   f(X) -> n__f(X),
                 true() -> n__true(),
            activate(X) -> X,
      activate(n__f(X)) -> f(activate(X)),
    activate(n__true()) -> true()}
  Weak:
   {}
  DP:
   Strict:
    {               f#(X) -> if#(X, c(), n__f(n__true())),
       activate#(n__f(X)) -> f#(activate(X)),
       activate#(n__f(X)) -> activate#(X),
     activate#(n__true()) -> true#()}
   Weak:
   {   if(true(), X, Y) -> X,
                   f(X) -> if(X, c(), n__f(n__true())),
                   f(X) -> n__f(X),
                 true() -> n__true(),
            activate(X) -> X,
      activate(n__f(X)) -> f(activate(X)),
    activate(n__true()) -> true()}
   EDG:
    {(activate#(n__f(X)) -> f#(activate(X)), f#(X) -> if#(X, c(), n__f(n__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)) -> activate#(X), activate#(n__true()) -> true#())}
    SCCS:
     Scc:
      {activate#(n__f(X)) -> activate#(X)}
     SCC:
      Strict:
       {activate#(n__f(X)) -> activate#(X)}
      Weak:
      {   if(true(), X, Y) -> X,
                      f(X) -> if(X, c(), n__f(n__true())),
                      f(X) -> n__f(X),
                    true() -> n__true(),
               activate(X) -> X,
         activate(n__f(X)) -> f(activate(X)),
       activate(n__true()) -> true()}
      SPSC:
       Simple Projection:
        pi(activate#) = 0
       Strict:
        {}
       Qed