Definition nat_to_pos (n : nat) : (n <> 0) -> positive :=
  fun _ => (P_of_succ_nat (pred n)).

Lemma factor_s_conv : forall(a : nat)(pf1 : (a <> 0))(pf2 : (S a) <> 0), (nat_to_pos (pf2)) = (Pos.succ (nat_to_pos pf1)).