helm/software/matita/contribs/RELATIONAL/NLE/props.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 set "baseuri" "cic:/matita/RELATIONAL/NLE/props".
  16
  17 include "NLE/fwd.ma".
  18
  19 theorem nle_refl: \forall x. x <= x.
  20  intros 1; elim x; clear x; intros; auto.
  21 qed.
  22
  23 theorem nle_trans_succ: \forall x,y. x <= y \to x <= succ y.
  24  intros. elim H; clear H x y; intros; auto.
  25 qed.
  26
  27 theorem nle_lt_or_eq: \forall y,x. x <= y \to x < y \lor x = y.
  28  intros 1. elim y; clear y; intros;
  29  [ lapply linear nle_gen_zero_2 to H. auto
  30  | lapply linear nle_gen_succ_2 to H1. decompose;
  31    [ rewrite > H1. clear H1. auto
  32    | lapply linear H to H3 as H0. decompose;
  33      [ rewrite > H1. clear H1 x. auto
  34      | rewrite < H. clear H n. auto
  35      ]
  36    ]
  37  ].
  38 qed.