matita/contribs/RELATIONAL/NLE/order.ma

   1 (**************************************************************************)
   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: \forall x. x <= x.
  20  intros; elim x; clear x; autobatch.
  21 qed.
  22
  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;
  26  [ autobatch
  27  | lapply linear nle_inv_succ_1 to H3. decompose. destruct.
  28    autobatch
  29  ].
  30 qed.
  31
  32 theorem nle_false: \forall x,y. x <= y \to y < x \to False.
  33  intros 3; elim H; clear H x y; autobatch.
  34 qed.
  35
  36 theorem nle_irrefl: \forall x. x < x \to False.
  37  intros. autobatch.
  38 qed.
  39
  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.
  42 qed.
  43
  44 theorem nle_irrefl_smart: \forall x. x < x \to False.
  45  intros 1. elim x; clear x; autobatch.
  46 qed.
  47
  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;
  50  [ elim n; clear n
  51  | decompose
  52  ]; autobatch.
  53 qed.