(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/RELATIONAL/NLE/inv".
+
include "NLE/defs.ma".
-theorem nle_inv_succ_1: \forall x,y. x < y \to
- \exists z. y = succ z \land x <= z.
- intros. inversion H; clear H; intros; subst. autobatch.
+theorem nle_inv_succ_1: ∀x,y. x < y →
+ ∃z. y = succ z ∧ x ≤ z.
+ intros; inversion H; clear H; intros; destruct. autobatch.
qed.
-theorem nle_inv_succ_succ: \forall x,y. x < succ y \to x <= y.
- intros. inversion H; clear H; intros; subst. autobatch.
+theorem nle_inv_succ_succ: ∀x,y. x < succ y → x ≤ y.
+ intros; inversion H; clear H; intros; destruct. autobatch.
qed.
-theorem nle_inv_succ_zero: \forall x. x < zero \to False.
- intros. inversion H; clear H; intros; subst.
+theorem nle_inv_succ_zero: ∀x. x < zero → False.
+ intros. inversion H; clear H; intros; destruct.
qed.
-theorem nle_inv_zero_2: \forall x. x <= zero \to x = zero.
- intros. inversion H; clear H; intros; subst. autobatch.
+theorem nle_inv_zero_2: ∀x. x ≤ zero → x = zero.
+ intros; inversion H; clear H; intros; destruct. autobatch.
qed.
-theorem nle_inv_succ_2: \forall y,x. x <= succ y \to
- x = zero \lor \exists z. x = succ z \land z <= y.
- intros. inversion H; clear H; intros; subst; auto depth = 4.
+theorem nle_inv_succ_2: ∀y,x. x ≤ succ y →
+ x = zero ∨ ∃z. x = succ z ∧ z ≤ y.
+ intros; inversion H; clear H; intros; destruct;
+ autobatch depth = 4.
qed.