Volume 1, 1989

University of Bialystok

Copyright (c) 1989 Association of Mizar Users

### The abstract of the Mizar article:

### Boolean Domains

**by****Andrzej Trybulec, and****Agata Darmochwal**- Received April 14, 1989
- MML identifier: FINSUB_1

- [ Mizar article, MML identifier index ]

environ vocabulary BOOLE, FINSET_1, FINSUB_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, FINSET_1; constructors TARSKI, SUBSET_1, FINSET_1, XBOOLE_0; clusters FINSET_1, SUBSET_1, XBOOLE_0, ZFMISC_1; requirements SUBSET, BOOLE; begin :::::::::::::: Auxiliary theorems ::::::::::::::::::: reserve X,Y for set; definition let IT be set; attr IT is cup-closed means :: FINSUB_1:def 1 for X,Y being set st X in IT & Y in IT holds X \/ Y in IT; attr IT is cap-closed means :: FINSUB_1:def 2 for X,Y being set st X in IT & Y in IT holds X /\ Y in IT; attr IT is diff-closed means :: FINSUB_1:def 3 for X,Y being set st X in IT & Y in IT holds X \ Y in IT; end; definition let IT be set; attr IT is preBoolean means :: FINSUB_1:def 4 IT is cup-closed diff-closed; end; definition cluster preBoolean -> cup-closed diff-closed set; cluster cup-closed diff-closed -> preBoolean set; end; definition cluster non empty cup-closed cap-closed diff-closed set; end; reserve A for non empty preBoolean set; canceled 9; theorem :: FINSUB_1:10 for A being set holds A is preBoolean iff for X,Y being set st X in A & Y in A holds X \/ Y in A & X \ Y in A; definition let A; let X,Y be Element of A; redefine func X \/ Y -> Element of A; func X \ Y -> Element of A; end; canceled 2; theorem :: FINSUB_1:13 X is Element of A & Y is Element of A implies X /\ Y is Element of A; theorem :: FINSUB_1:14 X is Element of A & Y is Element of A implies X \+\ Y is Element of A; theorem :: FINSUB_1:15 for A being non empty set st for X,Y being Element of A holds X \+\ Y in A & X \ Y in A holds A is preBoolean; theorem :: FINSUB_1:16 for A being non empty set st for X,Y being Element of A holds X \+\ Y in A & X /\ Y in A holds A is preBoolean; theorem :: FINSUB_1:17 for A being non empty set st for X,Y being Element of A holds X \+\ Y in A & X \/ Y in A holds A is preBoolean; definition let A; let X,Y be Element of A; redefine func X /\ Y -> Element of A; func X \+\ Y -> Element of A; end; theorem :: FINSUB_1:18 {} in A; canceled; theorem :: FINSUB_1:20 for A being set holds bool A is preBoolean; theorem :: FINSUB_1:21 for A,B being non empty preBoolean set holds A /\ B is non empty preBoolean; reserve A,B,P for set; reserve x for set; :::::::::::::: The set of all finite subsets of a set :::::::::::::::: definition let A; func Fin A -> preBoolean set means :: FINSUB_1:def 5 for X being set holds X in it iff X c= A & X is finite; end; definition let A; cluster Fin A -> non empty; end; definition let A; cluster -> finite Element of Fin A; end; canceled; theorem :: FINSUB_1:23 A c= B implies Fin A c= Fin B; theorem :: FINSUB_1:24 Fin (A /\ B) = Fin A /\ Fin B; theorem :: FINSUB_1:25 Fin A \/ Fin B c= Fin (A \/ B); theorem :: FINSUB_1:26 Fin A c= bool A; theorem :: FINSUB_1:27 A is finite implies Fin A = bool A; theorem :: FINSUB_1:28 Fin {} = { {} }; :::::::::::::: Finite subsets :::::::::::::::: definition let A; mode Finite_Subset of A is Element of Fin A; end; canceled; theorem :: FINSUB_1:30 for X being Finite_Subset of A holds X is finite; canceled; theorem :: FINSUB_1:32 for X being Finite_Subset of A holds X is Subset of A; canceled; theorem :: FINSUB_1:34 A is finite implies for X being Subset of A holds X is Finite_Subset of A;

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