YES

TRS:

  if(true(),x,y) -> x
  if(false(),x,y) -> y
  if(x,y,y) -> y
  if(if(x,y,z),u(),v()) -> if(x,if(y,u(),v()),if(z,u(),v()))

linear polynomial interpretations on N:

  if_A(x1,x2,x3) = x1 + x2 + x3
  if#_A(x1,x2,x3) = x1 + x2 + x3
  true_A = 0
  true#_A = 0
  false_A = 0
  false#_A = 0
  u_A = 0
  u#_A = 0
  v_A = 0
  v#_A = 0

precedence:

  if = true = false > v > u