∀ {p : ℕ} {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Fact (Nat.Prime p)]
  (f : R →+* S) {x : WittVector p R}, ((WittVector.map f) x = 0) = ∀ (n : ℕ), f (x.coeff n) = 0