Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

### The abstract of the Mizar article:

### Complete Lattices

**by****Grzegorz Bancerek**- Received May 13, 1992
- MML identifier: LATTICE3

- [ Mizar article, MML identifier index ]

environ vocabulary LATTICES, FUNCT_1, BOOLE, BINOP_1, SUBSET_1, FILTER_1, ORDERS_1, RELAT_1, RELAT_2, BHSP_3, FILTER_0, TARSKI, ZF_LANG, LATTICE3, PARTFUN1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, DOMAIN_1, STRUCT_0, FUNCT_1, FUNCT_2, BINOP_1, LATTICES, FILTER_0, LATTICE2, FILTER_1, RELAT_1, RELAT_2, RELSET_1, PARTFUN1, ORDERS_1; constructors DOMAIN_1, BINOP_1, LATTICE2, FILTER_1, RELAT_2, ORDERS_1, ORDERS_2, MEMBERED, PARTFUN1, XBOOLE_0; clusters LATTICE2, ORDERS_1, RELSET_1, STRUCT_0, RLSUB_2, SUBSET_1, LATTICES, MEMBERED, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; begin definition let X be set; func BooleLatt X -> strict LattStr means :: LATTICE3:def 1 the carrier of it = bool X & for Y,Z being Element of bool X holds (the L_join of it).(Y,Z) = Y \/ Z & (the L_meet of it).(Y,Z) = Y /\ Z; end; reserve X for set, x,y,z for Element of BooleLatt X, s for set; definition let X be set; cluster BooleLatt X -> non empty; end; theorem :: LATTICE3:1 x "\/" y = x \/ y & x "/\" y = x /\ y; theorem :: LATTICE3:2 x [= y iff x c= y; definition let X; cluster BooleLatt X -> Lattice-like; end; reserve y for Element of BooleLatt X; theorem :: LATTICE3:3 BooleLatt X is lower-bounded & Bottom BooleLatt X = {}; theorem :: LATTICE3:4 BooleLatt X is upper-bounded & Top BooleLatt X = X; definition let X; cluster BooleLatt X -> Boolean Lattice-like; end; theorem :: LATTICE3:5 for x being Element of BooleLatt X holds x` = X \ x; begin :: Correspondence Between Lattices and Posets definition let L be Lattice; redefine func LattRel L -> Order of the carrier of L; end; definition let L be Lattice; func LattPOSet L -> strict Poset equals :: LATTICE3:def 2 RelStr(#the carrier of L, LattRel L#); end; definition let L be Lattice; cluster LattPOSet L -> non empty; end; theorem :: LATTICE3:6 for L1,L2 being Lattice st LattPOSet L1 = LattPOSet L2 holds the LattStr of L1 = the LattStr of L2; definition let L be Lattice, p be Element of L; func p% -> Element of LattPOSet L equals :: LATTICE3:def 3 p; end; definition let L be Lattice, p be Element of LattPOSet L; func %p -> Element of L equals :: LATTICE3:def 4 p; end; reserve L for Lattice, p,q for Element of L; theorem :: LATTICE3:7 p [= q iff p% <= q%; definition let X be set, O be Order of X; redefine func O~ -> Order of X; end; definition let A be RelStr; func A~ -> strict RelStr equals :: LATTICE3:def 5 RelStr(#the carrier of A, (the InternalRel of A)~#); end; definition let A be Poset; cluster A~ -> reflexive transitive antisymmetric; end; definition let A be non empty RelStr; cluster A~ -> non empty; end; reserve A for RelStr, a,b,c for Element of A; theorem :: LATTICE3:8 A~~ = the RelStr of A; definition let A be RelStr, a be Element of A; func a~ -> Element of A~ equals :: LATTICE3:def 6 a; end; definition let A be RelStr, a be Element of A~; func ~a -> Element of A equals :: LATTICE3:def 7 a; end; theorem :: LATTICE3:9 a <= b iff b~ <= a~; definition let A be RelStr, X be set, a be Element of A; pred a is_<=_than X means :: LATTICE3:def 8 for b being Element of A st b in X holds a <= b; synonym X is_>=_than a; pred X is_<=_than a means :: LATTICE3:def 9 for b being Element of A st b in X holds b <= a; synonym a is_>=_than X; end; definition let IT be RelStr; attr IT is with_suprema means :: LATTICE3:def 10 for x,y being Element of IT ex z being Element of IT st x <= z & y <= z & for z' being Element of IT st x <= z' & y <= z' holds z <= z'; attr IT is with_infima means :: LATTICE3:def 11 for x,y being Element of IT ex z being Element of IT st z <= x & z <= y & for z' being Element of IT st z' <= x & z' <= y holds z' <= z; end; definition cluster with_suprema -> non empty RelStr; cluster with_infima -> non empty RelStr; end; theorem :: LATTICE3:10 A is with_suprema iff A~ is with_infima; theorem :: LATTICE3:11 for L being Lattice holds LattPOSet L is with_suprema with_infima; definition let IT be RelStr; attr IT is complete means :: LATTICE3:def 12 for X being set ex a being Element of IT st X is_<=_than a & for b being Element of IT st X is_<=_than b holds a <= b; end; definition cluster strict complete non empty Poset; end; reserve A for non empty RelStr, a,b,c,c' for Element of A; theorem :: LATTICE3:12 A is complete implies A is with_suprema with_infima; definition cluster complete with_suprema with_infima strict non empty Poset; end; definition let A be RelStr such that A is antisymmetric; let a,b be Element of A such that ex x being Element of A st a <= x & b <= x & for c being Element of A st a <= c & b <= c holds x <= c; func a"\/"b -> Element of A means :: LATTICE3:def 13 a <= it & b <= it & for c being Element of A st a <= c & b <= c holds it <= c; end; definition let A be RelStr such that A is antisymmetric; let a,b be Element of A such that ex x being Element of A st a >= x & b >= x & for c being Element of A st a >= c & b >= c holds x >= c; func a"/\"b -> Element of A means :: LATTICE3:def 14 it <= a & it <= b & for c being Element of A st c <= a & c <= b holds c <= it; end; reserve V for with_suprema antisymmetric RelStr, u1,u2,u3,u4 for Element of V; reserve N for with_infima antisymmetric RelStr, n1,n2,n3,n4 for Element of N; reserve K for with_suprema with_infima reflexive antisymmetric RelStr, k1,k2,k3 for Element of K; theorem :: LATTICE3:13 u1 "\/" u2 = u2 "\/" u1; theorem :: LATTICE3:14 V is transitive implies (u1 "\/" u2) "\/" u3 = u1 "\/" (u2 "\/" u3); theorem :: LATTICE3:15 n1 "/\" n2 = n2 "/\" n1; theorem :: LATTICE3:16 N is transitive implies (n1 "/\" n2) "/\" n3 = n1 "/\" (n2 "/\" n3); definition let L be with_infima antisymmetric RelStr, x, y be Element of L; redefine func x "/\" y; commutativity; end; definition let L be with_suprema antisymmetric RelStr, x, y be Element of L; redefine func x "\/" y; commutativity; end; theorem :: LATTICE3:17 (k1 "/\" k2) "\/" k2 = k2; theorem :: LATTICE3:18 k1 "/\" (k1 "\/" k2) = k1; theorem :: LATTICE3:19 for A being with_suprema with_infima Poset ex L being strict Lattice st the RelStr of A = LattPOSet L; definition let A be RelStr such that A is with_suprema with_infima Poset; func latt A -> strict Lattice means :: LATTICE3:def 15 the RelStr of A = LattPOSet it; end; theorem :: LATTICE3:20 (LattRel L)~ = LattRel (L.:) & (LattPOSet L)~ = LattPOSet (L.:); begin :: Complete Lattice definition let L be non empty LattStr, p be Element of L, X be set; pred p is_less_than X means :: LATTICE3:def 16 for q being Element of L st q in X holds p [= q; synonym X is_great_than p; pred X is_less_than p means :: LATTICE3:def 17 for q being Element of L st q in X holds q [= p; synonym p is_great_than X; end; theorem :: LATTICE3:21 for L being Lattice, p,q,r being Element of L holds p is_less_than {q,r} iff p [= q"/\"r; theorem :: LATTICE3:22 for L being Lattice, p,q,r being Element of L holds p is_great_than {q,r} iff q"\/"r [= p; definition let IT be non empty LattStr; attr IT is complete means :: LATTICE3:def 18 for X being set ex p being Element of IT st X is_less_than p & for r being Element of IT st X is_less_than r holds p [= r; attr IT is \/-distributive means :: LATTICE3:def 19 for X for a,b,c being Element of IT st X is_less_than a & (for d being Element of IT st X is_less_than d holds a [= d) & {b"/\"a' where a' is Element of IT: a' in X} is_less_than c & for d being Element of IT st {b"/\"a' where a' is Element of IT: a' in X} is_less_than d holds c [= d holds b"/\"a [= c; attr IT is /\-distributive means :: LATTICE3:def 20 for X for a,b,c being Element of IT st X is_great_than a & (for d being Element of IT st X is_great_than d holds d [= a) & {b"\/"a' where a' is Element of IT: a' in X} is_great_than c & for d being Element of IT st {b"\/"a' where a' is Element of IT: a' in X} is_great_than d holds d [= c holds c [= b"\/"a; end; theorem :: LATTICE3:23 for B being B_Lattice, a being Element of B holds X is_less_than a iff {b` where b is Element of B: b in X} is_great_than a`; theorem :: LATTICE3:24 for B being B_Lattice, a being Element of B holds X is_great_than a iff {b` where b is Element of B: b in X} is_less_than a`; theorem :: LATTICE3:25 BooleLatt X is complete; theorem :: LATTICE3:26 BooleLatt X is \/-distributive; theorem :: LATTICE3:27 BooleLatt X is /\-distributive; definition cluster complete \/-distributive /\-distributive strict Lattice; end; reserve p',q' for Element of LattPOSet L; theorem :: LATTICE3:28 p is_less_than X iff p% is_<=_than X; theorem :: LATTICE3:29 p' is_<=_than X iff %p' is_less_than X; theorem :: LATTICE3:30 X is_less_than p iff X is_<=_than p%; theorem :: LATTICE3:31 X is_<=_than p' iff X is_less_than %p'; definition let A be complete (non empty Poset); cluster latt A -> complete; end; definition let L be non empty LattStr such that L is complete Lattice; let X be set; func "\/"(X,L) -> Element of L means :: LATTICE3:def 21 X is_less_than it & for r being Element of L st X is_less_than r holds it [= r; end; definition let L be non empty LattStr, X be set; func "/\"(X,L) -> Element of L equals :: LATTICE3:def 22 "\/"({p where p is Element of L: p is_less_than X},L); end; definition let L be non empty LattStr, X be Subset of L; redefine func "\/"(X,L); synonym "\/" X; func "/\"(X,L); synonym "/\" X; end; reserve C for complete Lattice, a,a',b,b',c,d for Element of C, X,Y for set; theorem :: LATTICE3:32 "\/"({a"/\"b: b in X}, C) [= a"/\""\/"(X,C); theorem :: LATTICE3:33 C is \/-distributive iff for X, a holds a"/\""\/"(X,C) [= "\/"({a"/\" b: b in X}, C); theorem :: LATTICE3:34 a = "/\"(X,C) iff a is_less_than X & for b st b is_less_than X holds b [= a; theorem :: LATTICE3:35 a"\/""/\"(X,C) [= "/\"({a"\/"b: b in X}, C); theorem :: LATTICE3:36 C is /\-distributive iff for X, a holds "/\"({a"\/"b: b in X}, C) [= a"\/""/\" (X,C); theorem :: LATTICE3:37 "\/"(X,C) = "/\"({a: a is_great_than X}, C); theorem :: LATTICE3:38 a in X implies a [= "\/"(X,C) & "/\"(X,C) [= a; canceled; theorem :: LATTICE3:40 a is_less_than X implies a [= "/\"(X,C); theorem :: LATTICE3:41 a in X & X is_less_than a implies "\/"(X,C) = a; theorem :: LATTICE3:42 a in X & a is_less_than X implies "/\"(X,C) = a; theorem :: LATTICE3:43 "\/"{a} = a & "/\"{a} = a; theorem :: LATTICE3:44 a"\/"b = "\/"{a,b} & a"/\"b = "/\"{a,b}; theorem :: LATTICE3:45 a = "\/"({b: b [= a}, C) & a = "/\"({c: a [= c}, C); theorem :: LATTICE3:46 X c= Y implies "\/"(X,C) [= "\/"(Y,C) & "/\"(Y,C) [= "/\"(X,C); theorem :: LATTICE3:47 "\/"(X,C) = "\/"({a: ex b st a [= b & b in X}, C) & "/\"(X,C) = "/\"({b: ex a st a [= b & a in X}, C); theorem :: LATTICE3:48 (for a st a in X ex b st a [= b & b in Y) implies "\/"(X,C) [= "\/"(Y,C); theorem :: LATTICE3:49 X c= bool the carrier of C implies "\/"(union X, C) = "\/"({"\/" Y where Y is Subset of C: Y in X}, C); theorem :: LATTICE3:50 C is 0_Lattice & Bottom C = "\/"({},C); theorem :: LATTICE3:51 C is 1_Lattice & Top C = "\/"(the carrier of C, C); theorem :: LATTICE3:52 C is I_Lattice implies a => b = "\/"({c: a"/\"c [= b}, C); theorem :: LATTICE3:53 C is I_Lattice implies C is \/-distributive; theorem :: LATTICE3:54 for D being complete \/-distributive Lattice, a being Element of D holds a "/\" "\/"(X,D) = "\/"({a"/\" b1 where b1 is Element of D: b1 in X}, D) & "\/"(X,D) "/\" a = "\/"({b2"/\" a where b2 is Element of D: b2 in X}, D); theorem :: LATTICE3:55 for D being complete /\-distributive Lattice, a being Element of D holds a "\/" "/\"(X,D) = "/\"({a"\/"b1 where b1 is Element of D: b1 in X}, D) & "/\"(X,D) "\/" a = "/\"({b2"\/" a where b2 is Element of D: b2 in X}, D); scheme SingleFraenkel{A()->set, B()->non empty set, P[set]}: {A() where a is Element of B(): P[a]} = {A()} provided ex a being Element of B() st P[a]; scheme FuncFraenkel{B()->non empty set, C()->non empty set, A(set)->Element of C(),f()->Function, P[set]}: f().:{A(x) where x is Element of B(): P[x]} = {f().A(x) where x is Element of B(): P[x]} provided C() c= dom f(); theorem :: LATTICE3:56 for D being non empty set, f being Function of bool D, D st (for a being Element of D holds f.{a} = a) & for X being Subset of bool D holds f.(f.:X) = f.(union X) ex L being complete strict Lattice st the carrier of L = D & for X being Subset of L holds "\/" X = f.X;

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