helm/software/matita/nlibrary/sets/sets.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 include "logic/connectives.ma".
  16
  17 nrecord powerset (A: Type) : Type[1] ≝ { mem: A → CProp }.
  18 (* This is a projection! *)
  19 ndefinition mem ≝ λA.λr:powerset A.match r with [mk_powerset f ⇒ f].
  20
  21 interpretation "powerset" 'powerset A = (powerset A).
  22
  23 interpretation "subset construction" 'subset \eta.x = (mk_powerset ? x).
  24
  25 interpretation "mem" 'mem a S = (mem ? S a).
  26
  27 ndefinition subseteq ≝ λA.λU,V.∀a:A. a ∈ U → a ∈ V.
  28
  29 interpretation "subseteq" 'subseteq U V = (subseteq ? U V).
  30
  31 ntheorem subseteq_refl: ∀A.∀S:Ω \sup A.S ⊆ S.
  32  #A; #S; #x; #H; nassumption;
  33 nqed.
  34
  35 ntheorem subseteq_trans: ∀A.∀S1,S2,S3: Ω \sup A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
  36  #A; #S1; #S2; #S3; #H12; #H23; #x; #H;
  37  napply (H23 ??); napply (H12 ??); nassumption;
  38 nqed.
  39
  40 ndefinition overlaps ≝ λA.λU,V:Ω \sup A.∃x:A.x ∈ U ∧ x ∈ V.
  41
  42 interpretation "overlaps" 'overlaps U V = (overlaps ? U V).
  43
  44 ndefinition intersects ≝ λA.λU,V:Ω \sup A. {x | x ∈ U ∧ x ∈ V }.
  45
  46 interpretation "intersects" 'intersects U V = (intersects ? U V).
  47
  48 ndefinition union ≝ λA.λU,V:Ω \sup A. {x | x ∈ U ∨ x ∈ V }.
  49
  50 interpretation "union" 'union U V = (union ? U V).
  51
  52 (*
  53 ndefinition singleton ≝ λA.λa:A.{b | a=b}.
  54
  55 interpretation "singleton" 'singl a = (singleton ? a).
  56 *)