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)
   SCC Processor:
    #sccs: 1
    #rules: 4
    #arcs: 8/25
    DPs:
     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)
    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
    CDG Processor:
     DPs:
      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)
     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)) -> 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))
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 5/16
      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
      Subterm Criterion Processor:
       simple projection:
        pi(activate#) = 0
       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