Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

### The abstract of the Mizar article:

### Lattice of Fuzzy Sets

**by****Takashi Mitsuishi, and****Grzegorz Bancerek**- Received August 12, 2003
- MML identifier: LFUZZY_0

- [ Mizar article, MML identifier index ]

environ vocabulary LFUZZY_0, ORDERS_1, ARYTM, RELAT_2, LATTICES, RCOMP_1, FUNCT_1, RELAT_1, QC_LANG1, FUZZY_3, SEQ_1, TARSKI, FUZZY_1, GROUP_1, LATTICE2, LATTICE3, WAYBEL_0, MEASURE5, SQUARE_1, BOOLE, YELLOW_0, SEQ_2, INTEGRA1, SEQ_4, PRE_TOPC, BHSP_3, ORDINAL2, YELLOW_1, PBOOLE, FUNCT_2, PARTFUN1, MONOID_0, CARD_3, WAYBEL_3, WAYBEL_1, FUNCOP_1; notation TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, FUNCT_1, FUNCT_2, DOMAIN_1, ORDINAL1, PARTFUN1, SUBSET_1, NUMBERS, XREAL_0, REAL_1, SQUARE_1, SEQ_1, SEQ_4, RCOMP_1, RELSET_1, STRUCT_0, ORDERS_1, PRE_TOPC, PBOOLE, MONOID_0, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, WAYBEL_1, WAYBEL_3, FUZZY_1, FUZZY_3, FUZZY_4, INTEGRA1, PRE_CIRC; constructors XCMPLX_0, SQUARE_1, INTEGRA1, WAYBEL_2, YELLOW10, PSCOMP_1, WAYBEL_3, FUZZY_4, DOMAIN_1, ORDERS_3, PRE_CIRC, MONOID_0; clusters SUBSET_1, LATTICE3, XREAL_0, YELLOW14, RELSET_1, ORDERS_4, STRUCT_0, MONOID_0, WAYBEL_2, WAYBEL10, WAYBEL17, WAYBEL_3, MEMBERED; requirements BOOLE, SUBSET, NUMERALS, REAL; begin :: Posets of Real Numbers definition let R be RelStr; attr R is real means :: LFUZZY_0:def 1 the carrier of R c= REAL & for x,y being real number st x in the carrier of R & y in the carrier of R holds [x,y] in the InternalRel of R iff x <= y; end; definition let R be RelStr; attr R is interval means :: LFUZZY_0:def 2 R is real & ex a,b being real number st a <= b & the carrier of R = [.a,b.]; end; definition cluster interval -> real non empty RelStr; end; definition cluster empty -> real RelStr; end; theorem :: LFUZZY_0:1 for X being Subset of REAL ex R being strict RelStr st the carrier of R = X & R is real; definition cluster interval strict RelStr; end; theorem :: LFUZZY_0:2 for R1,R2 being real RelStr st the carrier of R1 = the carrier of R2 holds the RelStr of R1 = the RelStr of R2; definition let R be non empty real RelStr; cluster -> real Element of R; end; definition let X be Subset of REAL; func RealPoset X -> real strict RelStr means :: LFUZZY_0:def 3 the carrier of it = X; end; definition let X be non empty Subset of REAL; cluster RealPoset X -> non empty; end; definition let R be RelStr; let x,y be Element of R; redefine pred x <= y; synonym x <<= y; synonym y >>= x; antonym x ~<= y; antonym y ~>= x; end; definition let x,y be real number; redefine pred x <= y; synonym x <R= y; antonym y <R x; synonym y >R= x; antonym x >R y; end; theorem :: LFUZZY_0:3 for R being non empty real RelStr for x,y being Element of R holds x <R= y iff x <<= y; definition cluster real -> reflexive antisymmetric transitive RelStr; end; definition cluster -> connected (real non empty RelStr); end; definition let R be non empty real RelStr; let x,y be Element of R; redefine func max(x,y) -> Element of R; end; definition let R be non empty real RelStr; let x,y be Element of R; redefine func min(x,y) -> Element of R; end; definition cluster -> with_suprema with_infima (real non empty RelStr); end; reserve X for non empty Subset of REAL, x,y,z,w for real number, R for real non empty RelStr, a,b,c for Element of R; theorem :: LFUZZY_0:4 a"\/"b = max(a,b); theorem :: LFUZZY_0:5 a"/\"b = min(a,b); theorem :: LFUZZY_0:6 (ex x st x in the carrier of R & for y st y in the carrier of R holds x <= y) iff R is lower-bounded; theorem :: LFUZZY_0:7 (ex x st x in the carrier of R & for y st y in the carrier of R holds x >= y) iff R is upper-bounded; definition cluster interval -> bounded (non empty RelStr); end; theorem :: LFUZZY_0:8 for R being interval non empty RelStr, X being set holds ex_sup_of X,R; definition cluster -> complete (interval non empty RelStr); end; definition cluster -> distributive Chain; end; definition cluster -> Heyting (interval non empty RelStr); end; definition cluster [.0,1 .] -> non empty; end; definition cluster RealPoset [.0,1 .] -> interval; end; begin :: Product of Heyting Lattices theorem :: LFUZZY_0:9 for I being non empty set for J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I st for i being Element of I holds J.i is sup-Semilattice holds product J is with_suprema; theorem :: LFUZZY_0:10 for I being non empty set for J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I st for i being Element of I holds J.i is Semilattice holds product J is with_infima; theorem :: LFUZZY_0:11 for I being non empty set for J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I st for i being Element of I holds J.i is Semilattice for f,g being Element of product J, i being Element of I holds (f "/\" g).i = (f.i) "/\" (g.i); theorem :: LFUZZY_0:12 for I being non empty set for J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I st for i being Element of I holds J.i is sup-Semilattice for f,g being Element of product J, i being Element of I holds (f "\/" g).i = (f.i) "\/" (g.i); theorem :: LFUZZY_0:13 for I being non empty set for J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I st for i being Element of I holds J.i is Heyting complete LATTICE holds product J is complete Heyting; definition let A be non empty set; let R be complete Heyting LATTICE; cluster R |^ A -> Heyting; end; begin :: Lattice of Fuzzy Sets definition let A be non empty set; func FuzzyLattice A -> Heyting complete LATTICE equals :: LFUZZY_0:def 4 (RealPoset [. 0,1 .]) |^ A; end; theorem :: LFUZZY_0:14 for A being non empty set holds the carrier of FuzzyLattice A = Funcs(A, [. 0, 1 .]); definition let A be non empty set; cluster FuzzyLattice A -> constituted-Functions; end; theorem :: LFUZZY_0:15 for R being complete Heyting LATTICE, X being Subset of R, y be Element of R holds "\/"(X,R) "/\" y = "\/"({x "/\" y where x is Element of R: x in X},R); definition let X be non empty set; let a be Element of FuzzyLattice X; func @a -> Membership_Func of X equals :: LFUZZY_0:def 5 a; end; definition let X be non empty set; let f be Membership_Func of X; func (X,f)@ -> Element of FuzzyLattice X equals :: LFUZZY_0:def 6 f; end; definition let X be non empty set; let f be Membership_Func of X; let x be Element of X; redefine func f.x -> Element of RealPoset [. 0,1 .]; end; definition let X be non empty set; let f be Element of FuzzyLattice X; let x be Element of X; redefine func f.x -> Element of RealPoset [. 0,1 .]; end; reserve C for non empty set, c for Element of C, f,g for Membership_Func of C, s,t for Element of FuzzyLattice C; theorem :: LFUZZY_0:16 (for c holds f.c <R= g.c) iff (C,f)@ <<= (C,g)@; theorem :: LFUZZY_0:17 s <<= t iff for c holds @s.c <R= @t.c; theorem :: LFUZZY_0:18 max(f,g) = (C,f)@ "\/" (C,g)@; theorem :: LFUZZY_0:19 s "\/" t = max(@s, @t); theorem :: LFUZZY_0:20 min(f,g) = (C,f)@ "/\" (C,g)@; theorem :: LFUZZY_0:21 s "/\" t = min(@s, @t); begin :: Associativity of composition of fuzzy relations scheme SupDistributivity { L() -> complete LATTICE, X, Y() -> non empty set, F(set,set) -> Element of L(), P,Q[set]}: "\/"({ "\/"({F(x,y) where y is Element of Y(): Q[y]}, L()) where x is Element of X(): P[x] }, L()) = "\/"({F(x,y) where x is Element of X(), y is Element of Y(): P[x] & Q[y]}, L()); scheme SupDistributivity' { L() -> complete LATTICE, X, Y() -> non empty set, F(set,set) -> Element of L(), P,Q[set]}: "\/"({ "\/"({F(x,y) where x is Element of X(): P[x]},L()) where y is Element of Y(): Q[y] }, L()) = "\/"({F(x,y) where x is Element of X(), y is Element of Y(): P[x] & Q[y]}, L()); scheme FraenkelF'R'{ A() -> non empty set,B() -> non empty set, F1, F2(set,set) -> set, P[set,set] } : { F1(u1,v1) where u1 is (Element of A()), v1 is Element of B() : P[u1,v1] } = { F2(u2,v2) where u2 is (Element of A()), v2 is Element of B() : P[u2,v2] } provided for u being (Element of A()), v being Element of B() st P[u,v] holds F1(u,v) = F2(u,v); scheme FraenkelF6''R { A() -> non empty set, B() -> non empty set, F1, F2(set,set) -> set, P[set,set], Q[set,set] } : { F1(u1,v1) where u1 is (Element of A()), v1 is Element of B() : P[u1,v1] } = { F2(u2,v2) where u2 is (Element of A()), v2 is Element of B() : Q[u2,v2] } provided for u being (Element of A()), v being Element of B() holds P[u,v] iff Q[u,v] and for u being (Element of A()), v being Element of B() st P[u,v] holds F1(u,v) = F2(u,v); scheme SupCommutativity { L() -> complete LATTICE, X, Y() -> non empty set, F1, F2(set,set) -> Element of L(), P,Q[set]}: "\/"({ "\/"({F1(x,y) where y is Element of Y(): Q[y]}, L()) where x is Element of X(): P[x] }, L()) = "\/"({ "\/"({F2(x',y') where x' is Element of X(): P[x']}, L()) where y' is Element of Y(): Q[y'] }, L()) provided for x being Element of X(), y being Element of Y() st P[x] & Q[y] holds F1(x,y) = F2(x,y); theorem :: LFUZZY_0:22 for X,Y,Z being non empty set for R being RMembership_Func of X,Y for S being RMembership_Func of Y,Z for x being Element of X, z being Element of Z holds (R (#) S). [x,z] = "\/"({R. [x,y] "/\" S. [y,z] where y is Element of Y: not contradiction}, RealPoset [. 0,1 .]); theorem :: LFUZZY_0:23 for X,Y,Z,W being non empty set for R being RMembership_Func of X,Y for S being RMembership_Func of Y,Z for T being RMembership_Func of Z,W holds (R (#) S) (#) T = R (#) (S (#) T);

Back to top