proposition: Theorem SSSSev__even : forall n, ev (S (S (S (S n)))) -> ev n.
proof: Proof. Admitted. Theorem even5_nonsense : ev 5 -> 2 + 2 = 9. Proof. Admitted. Lemma ev_even_firsttry : forall n, ev n -> exists k, n = double k. Proof. intros n E. inversion E as [| n' E']. - exists 0. reflexivity. - simpl. assert (I : (exists k', n' = double k') -> (exists k, S (S n') = double k)). { intros [k' Hk']. rewrite Hk'. exists (S k'). reflexivity. } apply I. Admitted. Lemma ev_even : forall n, ev n -> exists k, n = double k. Proof. intros n E. induction E as [|n' E' IH]. - exists 0. reflexivity. - destruct IH as [k' Hk']. rewrite Hk'. exists (S k'). reflexivity. Qed.
imports: Logic