YES

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

linear polynomial interpretations on N:

  f_A(x1) = x1 + 3
  f#_A(x1) = x1 + 4
  if_A(x1,x2,x3) = x1 + x2 + x3
  if#_A(x1,x2,x3) = x1 + x3
  c_A = 0
  c#_A = 0
  n__f_A(x1) = x1 + 3
  n__f#_A(x1) = x1 + 3
  n__true_A = 0
  n__true#_A = 0
  true_A = 1
  true#_A = 1
  false_A = 3
  false#_A = 0
  activate_A(x1) = x1 + 1
  activate#_A(x1) = x1 + 2

precedence:

  if > activate > true > n__true > f = false > c = n__f