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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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19 theorem nle_refl: \forall x. x <= x.
20 intros; elim x; clear x; autobatch.
23 theorem nle_trans: \forall x,y. x <= y \to
24 \forall z. y <= z \to x <= z.
25 intros 3. elim H; clear H x y;
27 | lapply linear nle_inv_succ_1 to H3. decompose. destruct.
32 theorem nle_false: \forall x,y. x <= y \to y < x \to False.
33 intros 3; elim H; clear H x y; autobatch.
36 theorem nle_irrefl: \forall x. x < x \to False.
40 theorem nle_irrefl_ei: \forall x, z. z <= x \to z = succ x \to False.
41 intros 3. elim H; clear H x z; destruct. autobatch.
44 theorem nle_irrefl_smart: \forall x. x < x \to False.
45 intros 1. elim x; clear x; autobatch.
48 theorem nle_lt_or_eq: \forall y, x. x <= y \to x < y \lor x = y.
49 intros. elim H; clear H x y;