YES

TRS:

  f(0()) -> true()
  f(1()) -> false()
  f(s(x)) -> f(x)
  if(true(),s(x),s(y)) -> s(x)
  if(false(),s(x),s(y)) -> s(y)
  g(x,c(y)) -> c(g(x,y))
  g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(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) = 1
  if_A(x1,x2,x3) = x1
  if#_A(x1,x2,x3) = 2
  g_A(x1,x2) = x2 + 1
  g#_A(x1,x2) = x2 + 1
  c_A(x1) = x1 + 2
  c#_A(x1) = 0

precedence:

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