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_zero: \forall q. zero <= q.
  20  unfold NLE.
  21  intros. apply ex_intro; auto. (**)
  22 qed.
  23
  24 theorem nle_succ: \forall p,q. p <= q \to succ p <= succ q.
  25  unfold NLE.
  26  intros. decompose.
  27  apply ex_intro; [|auto] (**)
  28 qed.
  29
  30 theorem nle_refl: \forall x. x <= x.
  31  intros 1; elim x; clear x; intros; auto new timeout=100.
  32 qed.
  33
  34 theorem nle_trans_succ: \forall x,y. x <= y \to x <= succ y.
  35  intros 1. elim x; clear x; intros;
  36  [ auto new timeout=100.
  37  | lapply linear nle_gen_succ_1 to H1 as H0. decompose H0. subst.
  38    auto new timeout=100.
  39  ].
  40 qed.
  41
  42 theorem nle_lt_or_eq: \forall y,x. x <= y \to x < y \lor x = y.
  43  intros 1. elim y; clear y; intros;
  44  [ lapply linear nle_gen_zero_2 to H. auto new timeout=100
  45  | lapply linear nle_gen_succ_2 to H1. decompose;
  46    [ subst. auto new timeout=100
  47    | lapply linear H to H3 as H0. decompose;
  48      subst; auto new timeout=100
  49    ]
  50  ].
  51 qed.