(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/RELATIONAL/NLE/order".
+
include "NLE/inv.ma".
theorem nle_refl: \forall x. x <= x.
- intros; elim x; clear x; auto.
+ intros; elim x; clear x; autobatch.
qed.
theorem nle_trans: \forall x,y. x <= y \to
\forall z. y <= z \to x <= z.
intros 3. elim H; clear H x y;
- [ auto
- | lapply linear nle_inv_succ_1 to H3. decompose. subst.
- auto
+ [ autobatch
+ | lapply linear nle_inv_succ_1 to H3. decompose. destruct.
+ autobatch
].
qed.
theorem nle_false: \forall x,y. x <= y \to y < x \to False.
- intros 3; elim H; clear H x y; auto.
+ intros 3; elim H; clear H x y; autobatch.
qed.
theorem nle_irrefl: \forall x. x < x \to False.
- intros. auto.
+ intros. autobatch.
qed.
theorem nle_irrefl_ei: \forall x, z. z <= x \to z = succ x \to False.
- intros 3. elim H; clear H x z;
- [ destruct H1
- | destruct H3. clear H3. subst. auto.
- ].
+ intros 3. elim H; clear H x z; destruct. autobatch.
qed.
theorem nle_irrefl_smart: \forall x. x < x \to False.
- intros 1. elim x; clear x; auto.
+ intros 1. elim x; clear x; autobatch.
qed.
theorem nle_lt_or_eq: \forall y, x. x <= y \to x < y \lor x = y.
intros. elim H; clear H x y;
[ elim n; clear n
| decompose
- ]; auto.
+ ]; autobatch.
qed.