YES

TRS:

  f(0()) -> true()
  f(1()) -> false()
  f(s(x)) -> f(x)
  if(true(),x,y) -> x
  if(false(),x,y) -> y
  g(s(x),s(y)) -> if(f(x),s(x),s(y))
  g(x,c(y)) -> g(x,g(s(c(y)),y))

max/plus interpretations on N:

  f_A(x1) = max{1, -6}
  f#_A(x1) = max{1, -5 + x1}
  0_A = 1
  0#_A = 1
  true_A = 0
  true#_A = 0
  1_A = 1
  1#_A = 3
  false_A = 0
  false#_A = 2
  s_A(x1) = max{6, 1}
  s#_A(x1) = max{12, x1}
  if_A(x1,x2,x3) = max{0, 5 + x1, x2, x3}
  if#_A(x1,x2,x3) = max{12, 10, 12, 15}
  g_A(x1,x2) = max{6, -1, -1}
  g#_A(x1,x2) = max{13, 14, 10 + x2}
  c_A(x1) = max{10, 7 + x1}
  c#_A(x1) = max{8, 9}

precedence:

  c > 1 = g > 0 = false = s = if > f > true