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))

linear polynomial interpretations on N:

  f_A(x1) = 1
  f#_A(x1) = 1
  0_A = 1
  0#_A = 1
  true_A = 1
  true#_A = 0
  1_A = 1
  1#_A = 1
  false_A = 1
  false#_A = 0
  s_A(x1) = 1
  s#_A(x1) = x1
  if_A(x1,x2,x3) = x2 + x3
  if#_A(x1,x2,x3) = x1
  g_A(x1,x2) = 2
  g#_A(x1,x2) = x2 + 1
  c_A(x1) = x1 + 3
  c#_A(x1) = 0

precedence:

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