YES

TRS:

  fac(s(x)) -> *(fac(p(s(x))),s(x))
  p(s(0())) -> 0()
  p(s(s(x))) -> s(p(s(x)))

max/plus interpretations on N:

  fac_A(x1) = max{1, -1}
  fac#_A(x1) = max{15, 4 + x1}
  s_A(x1) = max{18, 12 + x1}
  s#_A(x1) = max{3, 1 + x1}
  *_A(x1,x2) = max{0, x1, 0}
  *#_A(x1,x2) = max{0, 14, 0}
  p_A(x1) = max{10, -8 + x1}
  p#_A(x1) = max{14, -14 + x1}
  0_A = 1
  0#_A = 3

precedence:

  fac > s > * = p > 0