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