TPTP Problem File: SET634+3.p
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- Solve Problem
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% File : SET634+3 : TPTP v8.2.0. Released v2.2.0.
% Domain : Set Theory
% Problem : X ^ (Y \ Z) = X ^ Y \ Z
% Version : [Try90] axioms : Reduced > Incomplete.
% English : The intersection of X and the difference of Y and Z is the
% intersection of X and the difference of Y 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 (116) [TS89]
% Status : Theorem
% Rating : 0.53 v8.2.0, 0.56 v7.5.0, 0.62 v7.4.0, 0.50 v7.3.0, 0.45 v7.2.0, 0.41 v7.1.0, 0.43 v6.4.0, 0.42 v6.3.0, 0.50 v6.2.0, 0.60 v6.1.0, 0.67 v6.0.0, 0.74 v5.4.0, 0.75 v5.3.0, 0.74 v5.2.0, 0.65 v5.1.0, 0.67 v5.0.0, 0.71 v4.1.0, 0.65 v4.0.0, 0.58 v3.7.0, 0.60 v3.5.0, 0.63 v3.4.0, 0.68 v3.3.0, 0.79 v3.2.0, 0.82 v3.1.0, 0.78 v2.7.0, 0.83 v2.6.0, 0.86 v2.5.0, 0.88 v2.4.0, 0.25 v2.3.0, 0.00 v2.2.1
% Syntax : Number of formulae : 9 ( 3 unt; 0 def)
% Number of atoms : 21 ( 5 equ)
% Maximal formula atoms : 3 ( 2 avg)
% Number of connectives : 13 ( 1 ~; 0 |; 3 &)
% ( 7 <=>; 2 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 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 : 23 ( 23 !; 0 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%--------------------------------------------------------------------------
%---- line(tarski - th(2),1832736)
fof(member_equal,axiom,
! [B,C] :
( ! [D] :
( member(D,B)
<=> member(D,C) )
=> B = 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(4),1833078)
fof(difference_defn,axiom,
! [B,C,D] :
( member(D,difference(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(intersection,2,function))
fof(commutativity_of_intersection,axiom,
! [B,C] : intersection(B,C) = intersection(C,B) ).
%---- line(hidden - axiom212,1832615)
fof(equal_member_defn,axiom,
! [B,C] :
( B = C
<=> ! [D] :
( member(D,B)
<=> member(D,C) ) ) ).
%---- 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(boole - th(116),1834426)
fof(prove_difference_and_intersection,conjecture,
! [B,C,D] : intersection(B,difference(C,D)) = difference(intersection(B,C),D) ).
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