helm/software/matita/nlibrary/sets/sets.ma

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   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/equality.ma".
  16
  17 nrecord powerset (A: Type) : Type[1] ≝ { mem: A → CProp }.
  18
  19 interpretation "powerset" 'powerset A = (powerset A).
  20
  21 interpretation "subset construction" 'subset \eta.x = (mk_powerset ? x).
  22
  23 interpretation "mem" 'mem a S = (mem ? S a).
  24
  25 ndefinition subseteq ≝ λA.λU,V.∀a:A. a ∈ U → a ∈ V.
  26
  27 interpretation "subseteq" 'subseteq U V = (subseteq ? U V).
  28
  29 ntheorem subseteq_refl: ∀A.∀S:Ω \sup A.S ⊆ S.
  30  #A; #S; #x; #H; nassumption;
  31 nqed.
  32
  33 ntheorem subseteq_trans: ∀A.∀S1,S2,S3: Ω \sup A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
  34  #A; #S1; #S2; #S3; #H12; #H23; #x; #H;
  35  napply (H23 …); napply (H12 …); nassumption;
  36 nqed.
  37
  38 ndefinition overlaps ≝ λA.λU,V:Ω \sup A.∃x:A.x ∈ U ∧ x ∈ V.
  39
  40 interpretation "overlaps" 'overlaps U V = (overlaps ? U V).
  41
  42 ndefinition intersects ≝ λA.λU,V:Ω \sup A. {x | x ∈ U ∧ x ∈ V }.
  43
  44 interpretation "intersects" 'intersects U V = (intersects ? U V).
  45
  46 ndefinition union ≝ λA.λU,V:Ω \sup A. {x | x ∈ U ∨ x ∈ V }.
  47
  48 interpretation "union" 'union U V = (union ? U V).
  49
  50 ndefinition singleton ≝ λA.λa:A.{b | a=b}.
  51
  52 interpretation "singleton" 'singl a = (singleton ? a).