proposition: Lemma e_is_irrational : forall (a : Z) (b : nat), b <> O -> e = (IZR a) / (INR b) -> False.
proof: Proof. intros [|a|a] b Hb He. refine (eirr O b Hb _). simpl in He; assumption. refine (eirr (nat_of_P a) b Hb _). rewrite INR_IPR. assumption. unfold IZR, e in *. assert (0 < IPR a / INR b). { rewrite <-INR_IPR. apply Rlt_mult_inv_pos. apply pos_INR_nat_of_P. INR_solve. } pose proof exp_pos 1 as Hpos. rewrite He in Hpos. lra. Qed.
imports: PeanoNat Rseries_facts Rsequence_facts Rsequence_subsequence Rpser_usual Reals Rsequence_tactics Rtactic Lra