TPTP Problem File: SET601+3.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SET601+3 : TPTP v9.0.0. Released v2.2.0.
% Domain : Set Theory
% Problem : X ^ Y U Y ^ Z U Z ^ X = (X U Y) ^ (Y U Z) ^ (Z U X)
% Version : [Try90] axioms : Reduced > Incomplete.
% English : The intersection of X and the union of Y and the intersection
% of Y and the union of Z and the intersection of Z and X is the
% intersection of (the union of X and Y) and the intersection of
% (the union of Y and Z) and (the union of Z and X).
% 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 (72) [TS89]
% Status : Theorem
% Rating : 0.36 v9.0.0, 0.39 v8.2.0, 0.36 v7.5.0, 0.38 v7.4.0, 0.33 v7.3.0, 0.24 v7.1.0, 0.26 v7.0.0, 0.33 v6.4.0, 0.35 v6.3.0, 0.33 v6.2.0, 0.40 v6.1.0, 0.47 v6.0.0, 0.43 v5.5.0, 0.52 v5.4.0, 0.57 v5.3.0, 0.63 v5.2.0, 0.50 v5.1.0, 0.48 v5.0.0, 0.50 v4.1.0, 0.39 v4.0.1, 0.43 v4.0.0, 0.50 v3.7.0, 0.45 v3.5.0, 0.42 v3.3.0, 0.29 v3.2.0, 0.27 v3.1.0, 0.22 v2.7.0, 0.50 v2.6.0, 0.43 v2.5.0, 0.38 v2.4.0, 0.50 v2.3.0, 0.67 v2.2.1
% Syntax : Number of formulae : 14 ( 9 unt; 0 def)
% Number of atoms : 24 ( 10 equ)
% Maximal formula atoms : 3 ( 1 avg)
% Number of connectives : 10 ( 0 ~; 1 |; 2 &)
% ( 6 <=>; 1 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 4 avg)
% Maximal term depth : 4 ( 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 : 34 ( 34 !; 0 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%------------------------------------------------------------------------------
%---- line(boole - th(64),1833712)
fof(associativity_of_union,axiom,
! [B,C,D] : union(union(B,C),D) = union(B,union(C,D)) ).
%---- line(boole - th(65),1833713)
fof(idempotency_of_intersection,axiom,
! [B] : intersection(B,B) = B ).
%---- line(boole - th(67),1833740)
fof(associativity_of_intersection,axiom,
! [B,C,D] : intersection(intersection(B,C),D) = intersection(B,intersection(C,D)) ).
%---- line(boole - th(69),1833784)
fof(union_intersection,axiom,
! [B,C] : union(B,intersection(B,C)) = B ).
%---- line(boole - th(71),1833842)
fof(union_distributes_over_intersection,axiom,
! [B,C,D] : union(B,intersection(C,D)) = intersection(union(B,C),union(B,D)) ).
%---- 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 - axiom118,1832615)
fof(equal_member_defn,axiom,
! [B,C] :
( B = C
<=> ! [D] :
( member(D,B)
<=> member(D,C) ) ) ).
%---- line(boole - th(72),1833851)
fof(prove_th72,conjecture,
! [B,C,D] : union(union(intersection(B,C),intersection(C,D)),intersection(D,B)) = intersection(intersection(union(B,C),union(C,D)),union(D,B)) ).
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