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

max/plus interpretations on N:

  f_A(x1) = max{7, 5 + x1}
  f#_A(x1) = max{10, 3 + x1}
  if_A(x1,x2,x3) = max{6, 6, x2, 2 + x3}
  if#_A(x1,x2,x3) = max{0, 2 + x1, 9 + x2, 4 + x3}
  c_A = 1
  c#_A = 9
  n__f_A(x1) = max{0, 5 + x1}
  n__f#_A(x1) = max{6, 2 + x1}
  n__true_A = 0
  n__true#_A = 0
  true_A = 2
  true#_A = 10
  false_A = 10
  false#_A = 0
  activate_A(x1) = max{1, 2 + x1}
  activate#_A(x1) = max{11, 1 + x1}

precedence:

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