%--------------------------------------------------------------------------
% File     : SET169+3 : TPTP v8.2.0. Released v2.2.0.
% Domain   : Set Theory
% Problem  : Intersection distributes over union
% Version  : [Try90] axioms : Reduced > Incomplete.
% English  : The intersection of X and (the union of Y and Z) is the union
%            of (the intersection of X and Y) and (the intersection of X
%            and Z).

% Refs     : [ILF] The ILF Group (1998), The ILF System: A Tool for the Int
%          : [Try90] Trybulec (1990), Tarski Grothendieck Set Theory
%          : [TS89]  Trybulec & Swieczkowska (1989), Boolean Properties of
% Source   : [ILF]
% Names    : BOOLE (70) [TS89]

% Status   : Theorem
% Rating   : 0.64 v8.2.0, 0.67 v8.1.0, 0.58 v7.5.0, 0.69 v7.4.0, 0.53 v7.3.0, 0.59 v7.2.0, 0.55 v7.1.0, 0.57 v7.0.0, 0.60 v6.4.0, 0.62 v6.3.0, 0.67 v6.2.0, 0.68 v6.1.0, 0.73 v6.0.0, 0.78 v5.4.0, 0.79 v5.3.0, 0.78 v5.2.0, 0.80 v5.1.0, 0.81 v5.0.0, 0.79 v4.1.0, 0.74 v4.0.0, 0.71 v3.7.0, 0.65 v3.5.0, 0.68 v3.3.0, 0.79 v3.2.0, 0.91 v3.1.0, 0.67 v2.7.0, 0.83 v2.6.0, 0.86 v2.5.0, 0.88 v2.4.0, 0.50 v2.3.0, 0.33 v2.2.1
% Syntax   : Number of formulae    :    9 (   4 unt;   0 def)
%            Number of atoms       :   19 (   5 equ)
%            Maximal formula atoms :    3 (   2 avg)
%            Number of connectives :   10 (   0   ~;   1   |;   2   &)
%                                         (   6 <=>;   1  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    6 (   5 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    3 (   2 usr;   0 prp; 2-2 aty)
%            Number of functors    :    2 (   2 usr;   0 con; 2-2 aty)
%            Number of variables   :   22 (  22   !;   0   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments :
%--------------------------------------------------------------------------
%---- line(boole - df(2),1833042)
fof(union_defn,axiom,
    ! [B,C,D] :
      ( member(D,union(B,C))
    <=> ( member(D,B)
        | member(D,C) ) ) ).

%---- line(boole - df(3),1833060)
fof(intersection_defn,axiom,
    ! [B,C,D] :
      ( member(D,intersection(B,C))
    <=> ( member(D,B)
        & member(D,C) ) ) ).

%---- line(boole - df(8),1833103)
fof(equal_defn,axiom,
    ! [B,C] :
      ( B = C
    <=> ( subset(B,C)
        & subset(C,B) ) ) ).

%---- property(commutativity,op(union,2,function))
fof(commutativity_of_union,axiom,
    ! [B,C] : union(B,C) = union(C,B) ).

%---- property(commutativity,op(intersection,2,function))
fof(commutativity_of_intersection,axiom,
    ! [B,C] : intersection(B,C) = intersection(C,B) ).

%---- line(tarski - df(3),1832749)
fof(subset_defn,axiom,
    ! [B,C] :
      ( subset(B,C)
    <=> ! [D] :
          ( member(D,B)
         => member(D,C) ) ) ).

%---- property(reflexivity,op(subset,2,predicate))
fof(reflexivity_of_subset,axiom,
    ! [B] : subset(B,B) ).

%---- line(hidden - axiom114,1832615)
fof(equal_member_defn,axiom,
    ! [B,C] :
      ( B = C
    <=> ! [D] :
          ( member(D,B)
        <=> member(D,C) ) ) ).

%---- line(boole - th(70),1833813)
fof(prove_intersection_distributes_over_union,conjecture,
    ! [B,C,D] : intersection(B,union(C,D)) = union(intersection(B,C),intersection(B,D)) ).

%--------------------------------------------------------------------------