matita/matita/contribs/RELATIONAL/NLE/order.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15
  16
  17 include "NLE/inv.ma".
  18
  19 theorem nle_refl: ∀x. x ≤ x.
  20  intros; elim x; clear x; autobatch.
  21 qed.
  22
  23 theorem nle_trans: ∀x,y. x ≤ y → ∀z. y ≤ z → x ≤ z.
  24  intros 3; elim H; clear H x y;
  25  [ autobatch
  26  | lapply linear nle_inv_succ_1 to H3. decompose. destruct.
  27    autobatch
  28  ].
  29 qed.
  30
  31 theorem nle_false: ∀x,y. x ≤ y → y < x → False.
  32  intros 3; elim H; clear H x y; autobatch.
  33 qed.
  34
  35 theorem nle_irrefl: ∀x. x < x → False.
  36  intros. autobatch.
  37 qed.
  38
  39 theorem nle_irrefl_ei: ∀x, z. z ≤ x → z = succ x → False.
  40  intros 3; elim H; clear H x z; destruct; autobatch.
  41 qed.
  42
  43 theorem nle_irrefl_smart: ∀x. x < x → False.
  44  intros 1. elim x; clear x; autobatch.
  45 qed.
  46
  47 theorem nle_lt_or_eq: ∀y, x. x ≤ y → x < y ∨ x = y.
  48  intros; elim H; clear H x y;
  49  [ elim n; clear n
  50  | decompose
  51  ]; autobatch.
  52 qed.