theorem test_inversion: \forall n. le n O \to n=O.
intros.
(* inversion begins *)
- cut O=O.
- (* goal 2: 0 = 0 *)
- goal 7. reflexivity.
- (* goal 1 *)
- generalize in match Hcut.
- apply (le_ind ? (\lambda x. O=x \to n=x) ? ? ? H).
- (* first goal (left open) *)
- intro. rewrite < H1.
- clear Hcut.
- (* second goal (closed) *)
- goal 14.
- simplify. intros.
- discriminate H3.
- (* inversion ends *)
+ cut (O=O);
+ [ 2: reflexivity;
+ | generalize in match Hcut.
+ apply (le_ind ? (\lambda x. O=x \to n=x) ? ? ? H);
+ [ intro. rewrite < H1. clear Hcut.
+ | simplify. intros. discriminate H3.
+ ]
reflexivity.
+ ]
qed.
(* Piu' semplice e non lascia l'ipotesi inutile Hcut *)
intros.
(* inversion begins *)
generalize in match (refl_equal nat O).
- apply (le_ind ? (\lambda x. O=x \to n=x) ? ? ? H).
- (* first goal (left open) *)
- intro. rewrite < H1.
- (* second goal (closed) *)
- goal 13.
- simplify. intros.
- discriminate H3.
- (* inversion ends *)
+ apply (le_ind ? (\lambda x. O=x \to n=x) ? ? ? H);
+ [ intro. rewrite < H1.
+ | simplify. intros. discriminate H3.
+ ]
reflexivity.
qed.