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)))

max/plus interpretations on N:

  f_A(x1) = max{2, 1}
  f#_A(x1) = max{0, 11}
  0_A = 1
  0#_A = 11
  true_A = 2
  true#_A = 10
  1_A = 1
  1#_A = 11
  false_A = 2
  false#_A = 10
  s_A(x1) = max{4, -10}
  s#_A(x1) = max{10, 2}
  if_A(x1,x2,x3) = max{4, -1, -14 + x2, 3}
  if#_A(x1,x2,x3) = max{10, 3, 3, 3}
  g_A(x1,x2) = max{3, -11, 1 + x2}
  g#_A(x1,x2) = max{5, 4, -1 + x2}
  c_A(x1) = max{13, 10 + x1}
  c#_A(x1) = max{6, 7 + x1}

precedence:

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