Volume 6, 1994

University of Bialystok

Copyright (c) 1994 Association of Mizar Users

### The abstract of the Mizar article:

### Boolean Properties of Lattices

**by****Agnieszka Julia Marasik**- Received March 28, 1994
- MML identifier: BOOLEALG

- [ Mizar article, MML identifier index ]

environ vocabulary LATTICES, BOOLE, SUBSET_1; notation STRUCT_0, LATTICES; constructors LATTICES, XBOOLE_0; clusters LATTICES, ZFMISC_1, XBOOLE_0; begin :: General Lattices reserve L for Lattice; reserve X,Y,Z,V for Element of L; definition let L,X,Y; func X \ Y -> Element of L equals :: BOOLEALG:def 1 X "/\" Y`; end; definition let L,X,Y; func X \+\ Y -> Element of L equals :: BOOLEALG:def 2 (X \ Y) "\/" (Y \ X); end; definition let L,X,Y; redefine pred X = Y means :: BOOLEALG:def 3 X [= Y & Y [= X; end; definition let L,X,Y; pred X meets Y means :: BOOLEALG:def 4 X "/\" Y <> Bottom L; antonym X misses Y; end; canceled 2; theorem :: BOOLEALG:3 X "\/" Y [= Z implies X [= Z & Y [= Z; theorem :: BOOLEALG:4 X "/\" Y [= X "\/" Z; canceled; theorem :: BOOLEALG:6 X [= Z implies X \ Y [= Z; theorem :: BOOLEALG:7 X [= Y implies X \ Z [= Y \ Z; theorem :: BOOLEALG:8 X \ Y [= X; theorem :: BOOLEALG:9 X \ Y [= X \+\ Y; theorem :: BOOLEALG:10 X \ Y [= Z & Y \ X [= Z implies X \+\ Y [= Z; theorem :: BOOLEALG:11 X = Y "\/" Z iff Y [= X & Z [= X & for V st Y [= V & Z [= V holds X [= V; theorem :: BOOLEALG:12 X = Y "/\" Z iff X [= Y & X [= Z & for V st V [= Y & V [= Z holds V [= X; canceled; theorem :: BOOLEALG:14 X "/\" (Y \ Z) = (X "/\" Y) \ Z; theorem :: BOOLEALG:15 X meets Y implies Y meets X; theorem :: BOOLEALG:16 X meets X iff X <> Bottom L; theorem :: BOOLEALG:17 X \+\ Y = Y \+\ X; definition let L, X, Y; redefine pred X meets Y; symmetry; redefine func X \+\ Y; commutativity; end; begin :: Modular Lattices reserve L for M_Lattice; reserve X,Y for Element of L; canceled 3; theorem :: BOOLEALG:21 X misses Y implies Y misses X; begin ::Distributive Lattices reserve L for D_Lattice; reserve X,Y,Z for Element of L; theorem :: BOOLEALG:22 (X "/\" Y) "\/" (X "/\" Z) = X implies X [= Y "\/" Z; canceled; theorem :: BOOLEALG:24 (X "\/" Y) \ Z = (X \ Z) "\/" (Y \ Z); begin ::Distributive bounded Lattices reserve L for 0_Lattice; reserve X,Y,Z for Element of L; theorem :: BOOLEALG:25 X [= Bottom L implies X = Bottom L; theorem :: BOOLEALG:26 X [= Y & X [= Z & Y "/\" Z = Bottom L implies X = Bottom L; theorem :: BOOLEALG:27 X "\/" Y = Bottom L iff X = Bottom L & Y = Bottom L; theorem :: BOOLEALG:28 X [= Y & Y "/\" Z = Bottom L implies X "/\" Z = Bottom L; theorem :: BOOLEALG:29 Bottom L \ X = Bottom L; theorem :: BOOLEALG:30 X meets Y & Y [= Z implies X meets Z; theorem :: BOOLEALG:31 X meets Y "/\" Z implies X meets Y & X meets Z; theorem :: BOOLEALG:32 X meets Y \ Z implies X meets Y; theorem :: BOOLEALG:33 X misses Bottom L; theorem :: BOOLEALG:34 X misses Z & Y [= Z implies X misses Y; theorem :: BOOLEALG:35 X misses Y or X misses Z implies X misses Y "/\" Z; theorem :: BOOLEALG:36 X [= Y & X [= Z & Y misses Z implies X = Bottom L; theorem :: BOOLEALG:37 X misses Y implies (Z "/\" X) misses (Z "/\" Y) & (X "/\" Z) misses (Y "/\" Z); begin ::Boolean Lattices reserve L for B_Lattice; reserve X,Y,Z,V for Element of L; theorem :: BOOLEALG:38 X \ Y [= Z implies X [= Y "\/" Z; theorem :: BOOLEALG:39 X [= Y implies Z \ Y [= Z \ X; theorem :: BOOLEALG:40 X [= Y & Z [= V implies X \ V [= Y \ Z; theorem :: BOOLEALG:41 X [= Y "\/" Z implies X \ Y [= Z & X \ Z [= Y; theorem :: BOOLEALG:42 X` [= (X "/\" Y)` & Y` [= (X "/\" Y)`; theorem :: BOOLEALG:43 (X "\/" Y)` [= X` & (X "\/" Y)` [= Y`; theorem :: BOOLEALG:44 X [= Y \ X implies X = Bottom L; theorem :: BOOLEALG:45 X [= Y implies Y = X "\/" (Y \ X); theorem :: BOOLEALG:46 X \ Y = Bottom L iff X [= Y; theorem :: BOOLEALG:47 X [= (Y "\/" Z) & X "/\" Z = Bottom L implies X [= Y; theorem :: BOOLEALG:48 X "\/" Y = (X \ Y) "\/" Y; theorem :: BOOLEALG:49 X \ (X "\/" Y) = Bottom L; theorem :: BOOLEALG:50 X \ X "/\" Y = X \ Y & X \ Y "/\" X = X \ Y; theorem :: BOOLEALG:51 (X \ Y) "/\" Y = Bottom L; theorem :: BOOLEALG:52 X "\/" (Y \ X) = X "\/" Y & (Y \ X) "\/" X = Y "\/" X; theorem :: BOOLEALG:53 (X "/\" Y) "\/" (X \ Y) = X; theorem :: BOOLEALG:54 X \ (Y \ Z) = (X \ Y) "\/" (X "/\" Z); theorem :: BOOLEALG:55 X \ (X \ Y) = X "/\" Y; theorem :: BOOLEALG:56 (X "\/" Y) \ Y = X \ Y; theorem :: BOOLEALG:57 X "/\" Y = Bottom L iff X \ Y = X; theorem :: BOOLEALG:58 X \ (Y "\/" Z) = (X \ Y) "/\" (X \ Z); theorem :: BOOLEALG:59 X \ (Y "/\" Z) = (X \ Y) "\/" (X \ Z); theorem :: BOOLEALG:60 X "/\" (Y \ Z) = X "/\" Y \ X "/\" Z & (Y \ Z) "/\" X = Y "/\" X \ Z "/\" X ; theorem :: BOOLEALG:61 (X "\/" Y) \ (X "/\" Y) = (X \ Y) "\/" (Y \ X); theorem :: BOOLEALG:62 (X \ Y) \ Z = X \ (Y "\/" Z); theorem :: BOOLEALG:63 X \ Y = Y \ X implies X = Y; canceled 2; theorem :: BOOLEALG:66 X \ X = Bottom L; theorem :: BOOLEALG:67 X \ Bottom L = X; theorem :: BOOLEALG:68 (X \ Y)` = X` "\/" Y; theorem :: BOOLEALG:69 X meets Y "\/" Z iff X meets Y or X meets Z; theorem :: BOOLEALG:70 X "/\" Y misses X \ Y; theorem :: BOOLEALG:71 X misses Y "\/" Z iff X misses Y & X misses Z; theorem :: BOOLEALG:72 (X \ Y) misses Y; theorem :: BOOLEALG:73 X misses Y implies (X "\/" Y) \ Y = X & (X "\/" Y) \ X = Y; theorem :: BOOLEALG:74 X` "\/" Y` = X "\/" Y & X misses X` & Y misses Y` implies X = Y` & Y = X`; theorem :: BOOLEALG:75 X` "\/" Y` = X "\/" Y & Y misses X` & X misses Y` implies X = X` & Y = Y`; theorem :: BOOLEALG:76 X \+\ Bottom L = X; theorem :: BOOLEALG:77 X \+\ X = Bottom L; theorem :: BOOLEALG:78 X "/\" Y misses X \+\ Y; theorem :: BOOLEALG:79 X "\/" Y = X \+\ (Y \ X); theorem :: BOOLEALG:80 X \+\ (X "/\" Y) = X \ Y; theorem :: BOOLEALG:81 X "\/" Y = (X \+\ Y) "\/" (X "/\" Y); theorem :: BOOLEALG:82 (X \+\ Y) \+\ (X "/\" Y) = X "\/" Y; theorem :: BOOLEALG:83 (X \+\ Y) \+\ (X "\/" Y) = X "/\" Y; theorem :: BOOLEALG:84 X \+\ Y = (X "\/" Y) \ (X "/\" Y); theorem :: BOOLEALG:85 (X \+\ Y) \ Z = (X \ (Y "\/" Z)) "\/" (Y \ (X "\/" Z)); theorem :: BOOLEALG:86 X \ (Y \+\ Z) = (X \ (Y "\/" Z)) "\/" (X "/\" Y "/\" Z); theorem :: BOOLEALG:87 (X \+\ Y) \+\ Z = X \+\ (Y \+\ Z); theorem :: BOOLEALG:88 (X \+\ Y)` = (X "/\" Y) "\/" (X` "/\" Y`);

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