### The Mizar article:

### Boolean Domains

**by****Andrzej Trybulec, and****Agata Darmochwal**

- Received April 14, 1989
Copyright (c) 1989 Association of Mizar Users

- MML identifier: FINSUB_1
- [ 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; definitions TARSKI, XBOOLE_0; theorems FINSET_1, TARSKI, ZFMISC_1, XBOOLE_0, XBOOLE_1; schemes XBOOLE_0; begin :::::::::::::: Auxiliary theorems ::::::::::::::::::: reserve X,Y for set; definition let IT be set; attr IT is cup-closed means :Def1: for X,Y being set st X in IT & Y in IT holds X \/ Y in IT; attr IT is cap-closed means for X,Y being set st X in IT & Y in IT holds X /\ Y in IT; attr IT is diff-closed means :Def3: 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 :Def4: IT is cup-closed diff-closed; end; definition cluster preBoolean -> cup-closed diff-closed set; coherence by Def4; cluster cup-closed diff-closed -> preBoolean set; coherence by Def4; end; definition cluster non empty cup-closed cap-closed diff-closed set; existence proof take {{}}; thus {{}} is non empty; thus {{}} is cup-closed proof let X,Y be set; assume X in {{}} & Y in {{}}; then X = {} & Y = {} by TARSKI:def 1; hence X \/ Y in {{}} by TARSKI:def 1; end; thus {{}} is cap-closed proof let X,Y be set; assume X in {{}} & Y in {{}}; then X = {} & Y = {} by TARSKI:def 1; hence X /\ Y in {{}} by TARSKI:def 1; end; thus {{}} is diff-closed proof let X,Y be set; assume X in {{}} & Y in {{}}; then X = {} & Y = {} by TARSKI:def 1; hence X \ Y in {{}} by TARSKI:def 1; end; end; end; reserve A for non empty preBoolean set; canceled 9; theorem Th10: 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 proof let A be set; hereby assume A is preBoolean; then A1: A is cup-closed diff-closed by Def4; let X,Y be set; assume X in A & Y in A; hence X \/ Y in A & X \ Y in A by A1,Def1,Def3; end; assume A2: for X,Y being set st X in A & Y in A holds X \/ Y in A & X \ Y in A; A3:A is cup-closed proof let X,Y be set; assume X in A & Y in A; hence X \/ Y in A by A2; end; A is diff-closed proof let X,Y be set; assume X in A & Y in A; hence X \ Y in A by A2; end; hence thesis by A3,Def4; end; definition let A; let X,Y be Element of A; redefine func X \/ Y -> Element of A; correctness by Th10; func X \ Y -> Element of A; correctness by Th10; end; canceled 2; theorem Th13: X is Element of A & Y is Element of A implies X /\ Y is Element of A proof assume X is Element of A & Y is Element of A; then reconsider X,Y as Element of A; X /\ Y = X \ (X \ Y) by XBOOLE_1:48; hence thesis; end; theorem Th14: X is Element of A & Y is Element of A implies X \+\ Y is Element of A proof assume X is Element of A & Y is Element of A; then reconsider X,Y as Element of A; X \+\ Y = (X \ Y) \/ (Y \ X) by XBOOLE_0:def 6; hence thesis; end; theorem 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 proof let A be non empty set such that A1: for X,Y being Element of A holds X \+\ Y in A & X \ Y in A; now let X,Y be set; assume A2: X in A & Y in A; then reconsider Z = Y \ X as Element of A by A1; X \/ Y = X \+\ Z by XBOOLE_1:98; hence X \/ Y in A by A1,A2; thus X \ Y in A by A1,A2; end; hence thesis by Th10; end; theorem 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 proof let A be non empty set such that A1: for X,Y being Element of A holds X \+\ Y in A & X /\ Y in A; now let X,Y be set; assume A2: X in A & Y in A; then reconsider Z1 = X \+\ Y, Z2 = X /\ Y as Element of A by A1; X \/ Y = Z1 \+\ Z2 by XBOOLE_1:94; hence X \/ Y in A by A1; X \ Y = X \+\ Z2 by XBOOLE_1:100; hence X \ Y in A by A1,A2; end; hence thesis by Th10; end; theorem 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 proof let A be non empty set such that A1: for X,Y being Element of A holds X \+\ Y in A & X \/ Y in A; now let X,Y be set; assume A2: X in A & Y in A; hence X \/ Y in A by A1; reconsider Z1 = X \+\ Y, Z2 = X \/ Y as Element of A by A1,A2; X /\ Y = Z1 \+\ Z2 by XBOOLE_1:95; then reconsider Z = X /\ Y as Element of A by A1; X \ Y = X \+\ Z by XBOOLE_1:100; hence X \ Y in A by A1,A2; end; hence thesis by Th10; end; definition let A; let X,Y be Element of A; redefine func X /\ Y -> Element of A; coherence by Th13; func X \+\ Y -> Element of A; coherence by Th14; end; theorem Th18: {} in A proof consider x being Element of A; x \ x = {} by XBOOLE_1:37; hence thesis; end; canceled; theorem for A being set holds bool A is preBoolean proof let A be set; now let X,Y be set; assume X in bool A & Y in bool A; then reconsider X'=X,Y'=Y as Element of bool A; X' \/ Y' in bool A & X' \ Y' in bool A; hence X \/ Y in bool A & X \ Y in bool A; end; hence thesis by Th10; end; theorem for A,B being non empty preBoolean set holds A /\ B is non empty preBoolean proof let A,B be non empty preBoolean set; {} in A & {} in B by Th18; then reconsider C = A /\ B as non empty set by XBOOLE_0:def 3; C is preBoolean proof now let X,Y be set; assume X in C & Y in C; then A1: X in A & Y in A & X in B & Y in B by XBOOLE_0:def 3; then X \/ Y in A & X \/ Y in B by Th10; hence X \/ Y in C by XBOOLE_0:def 3; X \ Y in A & X \ Y in B by A1,Th10; hence X \ Y in C by XBOOLE_0:def 3; end; hence thesis by Th10; end; hence thesis; end; 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 :Def5: for X being set holds X in it iff X c= A & X is finite; existence proof defpred P[set] means ex y being set st y=$1 & y c= A & y is finite; consider P such that A1:for x holds x in P iff x in bool A & P[x] from Separation; {} c= A by XBOOLE_1:2; then reconsider Q=P as non empty set by A1; for X,Y being set st X in Q & Y in Q holds X \/ Y in Q & X \ Y in Q proof let X,Y be set; assume A2: X in Q & Y in Q; then consider Z1 being set such that A3: Z1=X & Z1 c=A & Z1 is finite by A1; consider Z2 being set such that A4: Z2=Y & Z2 c= A & Z2 is finite by A1,A2; A5: Z1 \/ Z2 = X \/ Y & Z1 \/ Z2 c= A & Z1 \/ Z2 is finite by A3,A4,FINSET_1:14,XBOOLE_1:8; Z1 \ Z2 c= Z1 & Z1 \ Z2 is finite by A3,FINSET_1:16,XBOOLE_1:36; then Z1 \ Z2 = X \ Y & Z1 \ Z2 c= A & Z1 \ Z2 is finite by A3,A4,XBOOLE_1: 1; hence thesis by A1,A5; end; then reconsider R=Q as non empty preBoolean set by Th10; for X being set holds X in R iff X c= A & X is finite proof let X be set; thus X in R implies X c= A & X is finite proof assume X in R; then ex Y being set st Y=X & Y c= A & Y is finite by A1; hence thesis; end; thus (X c= A & X is finite) implies X in R by A1; end; hence thesis; end; uniqueness proof let P,Q be preBoolean set; assume that A6: for X being set holds X in P iff X c= A & X is finite and A7: for X being set holds X in Q iff X c= A & X is finite; for x being set holds x in P iff x in Q proof let x; thus x in P implies x in Q proof assume x in P; then x c= A & x is finite by A6; hence thesis by A7; end; thus x in Q implies x in P proof assume x in Q; then x c= A & x is finite by A7; hence thesis by A6; end; end; hence thesis by TARSKI:2; end; end; definition let A; cluster Fin A -> non empty; coherence proof {} c= A by XBOOLE_1:2; hence thesis by Def5; end; end; definition let A; cluster -> finite Element of Fin A; coherence by Def5; end; canceled; theorem Th23: A c= B implies Fin A c= Fin B proof assume A1: A c= B; now let X; assume X in Fin A; then X c= A & X is finite by Def5; then X c= B & X is finite by A1,XBOOLE_1:1; hence X in Fin B by Def5; end; hence thesis by TARSKI:def 3; end; theorem Fin (A /\ B) = Fin A /\ Fin B proof A /\ B c= A & A /\ B c= B by XBOOLE_1:17; then Fin (A /\ B) c= Fin A & Fin (A /\ B) c= Fin B by Th23; hence Fin (A /\ B) c= Fin A /\ Fin B by XBOOLE_1:19; now let X; assume X in Fin A /\ Fin B; then X in Fin A & X in Fin B by XBOOLE_0:def 3; then X c= A & X c= B & X is finite by Def5; then X c= A /\ B & X is finite by XBOOLE_1:19; hence X in Fin (A /\ B) by Def5; end; hence Fin A /\ Fin B c= Fin (A /\ B) by TARSKI:def 3; end; theorem Fin A \/ Fin B c= Fin (A \/ B) proof A c= A \/ B & B c= A \/ B by XBOOLE_1:7; then Fin A c= Fin (A \/ B) & Fin B c= Fin(A \/ B) by Th23; hence thesis by XBOOLE_1:8; end; theorem Th26: Fin A c= bool A proof let x; assume x in Fin A; then x c= A by Def5; hence thesis; end; theorem Th27: A is finite implies Fin A = bool A proof assume A1: A is finite; bool A c= Fin A proof let x; assume A2: x in bool A; then x is finite by A1,FINSET_1:13; hence thesis by A2,Def5; end; then bool A c= Fin A & Fin A c= bool A by Th26; hence thesis by XBOOLE_0:def 10; end; theorem Fin {} = { {} } by Th27,ZFMISC_1:1; :::::::::::::: Finite subsets :::::::::::::::: definition let A; mode Finite_Subset of A is Element of Fin A; end; canceled; theorem for X being Finite_Subset of A holds X is finite; canceled; theorem for X being Finite_Subset of A holds X is Subset of A by Def5; canceled; theorem A is finite implies for X being Subset of A holds X is Finite_Subset of A proof assume A1: A is finite; let X be Subset of A; X is finite by A1,FINSET_1:13; hence thesis by Def5; end;

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