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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: ∀x. x ≤ x.
20 intros; elim x; clear x; autobatch.
23 theorem nle_trans: ∀x,y. x ≤ y → ∀z. y ≤ z → x ≤ z.
24 intros 3; elim H; clear H x y;
26 | lapply linear nle_inv_succ_1 to H3. decompose. destruct.
31 theorem nle_false: ∀x,y. x ≤ y → y < x → False.
32 intros 3; elim H; clear H x y; autobatch.
35 theorem nle_irrefl: ∀x. x < x → False.
39 theorem nle_irrefl_ei: ∀x, z. z ≤ x → z = succ x → False.
40 intros 3; elim H; clear H x z; destruct; autobatch.
43 theorem nle_irrefl_smart: ∀x. x < x → False.
44 intros 1. elim x; clear x; autobatch.
47 theorem nle_lt_or_eq: ∀y, x. x ≤ y → x < y ∨ x = y.
48 intros; elim H; clear H x y;