FILTER_1: Filters - Part II. Quotient Lattices Modulo Filters and Direct Product of Two Lattices

:: Filters - Part II. Quotient Lattices Modulo Filters and Direct Product of Two Lattices
:: by Grzegorz Bancerek
::
:: Received April 19, 1991
:: Copyright (c) 1991-2011 Association of Mizar Users

begin

deffunc H₁( LattStr ) -> Element of bool [:[: the carrier of $1, the carrier of $1:], the carrier of $1:] = the L_join of $1;

deffunc H₂( LattStr ) -> Element of bool [:[: the carrier of $1, the carrier of $1:], the carrier of $1:] = the L_meet of $1;

theorem Th1: :: FILTER_1:1

for L being Lattice
for F1, F2 being Filter of L holds F1 /\ F2 is Filter of L

proof end;

theorem :: FILTER_1:2

for L being Lattice
for p, q being Element of L st <.p.) = <.q.) holds
p = q

proof end;

definition

let L be Lattice;
let F1, F2 be Filter of L;
:: original: /\
redefine func F1 /\ F2 -> Filter of L;
coherence by Th1;

end;

definition

let D be non empty set ;
let R be Relation;
mode UnOp of D,R -> UnOp of D means :Def1: :: FILTER_1:def 1
for x, y being Element of D st [x,y] in R holds
[(it . x),(it . y)] in R;
existence

proof end;

mode BinOp of D,R -> BinOp of D means :Def2: :: FILTER_1:def 2
for x1, y1, x2, y2 being Element of D st [x1,y1] in R & [x2,y2] in R holds
[(it . (x1,x2)),(it . (y1,y2))] in R;
existence

proof end;

end;

:: deftheorem Def1 defines UnOp FILTER_1:def 1 :

:: deftheorem Def2 defines BinOp FILTER_1:def 2 :

definition

let D be non empty set ;
let R be Equivalence_Relation of D;
mode UnOp of R is UnOp of D,R;
mode BinOp of R is BinOp of D,R;

end;

definition

let D be non empty set ;
let R be Equivalence_Relation of D;
let u be UnOp of D;
assume A1: u is UnOp of D,R ;
func u /\/ R -> UnOp of (Class R) means :: FILTER_1:def 3
for x, y being set st x in Class R & y in x holds
it . x = Class (R,(u . y));
existence

proof end;

proof end;

end;

:: deftheorem defines /\/ FILTER_1:def 3 :

definition

let D be non empty set ;
let R be Equivalence_Relation of D;
let b be BinOp of D;
assume A1: b is BinOp of D,R ;
func b /\/ R -> BinOp of (Class R) means :Def4: :: FILTER_1:def 4
for x, y, x1, y1 being set st x in Class R & y in Class R & x1 in x & y1 in y holds
it . (x,y) = Class (R,(b . (x1,y1)));
existence

proof end;

proof end;

end;

:: deftheorem Def4 defines /\/ FILTER_1:def 4 :

theorem Th3: :: FILTER_1:3

for D being non empty set
for RD being Equivalence_Relation of D
for a, b being Element of D
for F being BinOp of D,RD holds (F /\/ RD) . ((Class (RD,a)),(Class (RD,b))) = Class (RD,(F . (a,b)))

proof end;

scheme :: FILTER_1:sch 1
SchAux1{ F₁() -> non empty set , F₂() -> Equivalence_Relation of F₁(), P₁[ set ] } :

for x being Element of Class F₂() holds P₁[x]

provided

A1: for x being Element of F₁() holds P₁[ EqClass (F₂(),x)]

proof end;

scheme :: FILTER_1:sch 2
SchAux2{ F₁() -> non empty set , F₂() -> Equivalence_Relation of F₁(), P₁[ set , set ] } :

for x, y being Element of Class F₂() holds P₁[x,y]

provided

A1: for x, y being Element of F₁() holds P₁[ EqClass (F₂(),x), EqClass (F₂(),y)]

proof end;

scheme :: FILTER_1:sch 3
SchAux3{ F₁() -> non empty set , F₂() -> Equivalence_Relation of F₁(), P₁[ set , set , set ] } :

for x, y, z being Element of Class F₂() holds P₁[x,y,z]

provided

A1: for x, y, z being Element of F₁() holds P₁[ EqClass (F₂(),x), EqClass (F₂(),y), EqClass (F₂(),z)]

proof end;

theorem Th4: :: FILTER_1:4

for D being non empty set
for RD being Equivalence_Relation of D
for F being BinOp of D,RD st F is commutative holds
F /\/ RD is commutative

proof end;

theorem Th5: :: FILTER_1:5

for D being non empty set
for RD being Equivalence_Relation of D
for F being BinOp of D,RD st F is associative holds
F /\/ RD is associative

proof end;

theorem Th6: :: FILTER_1:6

for D being non empty set
for RD being Equivalence_Relation of D
for d being Element of D
for F being BinOp of D,RD st d is_a_left_unity_wrt F holds
EqClass (RD,d) is_a_left_unity_wrt F /\/ RD

proof end;

theorem Th7: :: FILTER_1:7

for D being non empty set
for RD being Equivalence_Relation of D
for d being Element of D
for F being BinOp of D,RD st d is_a_right_unity_wrt F holds
EqClass (RD,d) is_a_right_unity_wrt F /\/ RD

proof end;

theorem :: FILTER_1:8

for D being non empty set
for RD being Equivalence_Relation of D
for d being Element of D
for F being BinOp of D,RD st d is_a_unity_wrt F holds
EqClass (RD,d) is_a_unity_wrt F /\/ RD

proof end;

theorem Th9: :: FILTER_1:9

for D being non empty set
for RD being Equivalence_Relation of D
for F, G being BinOp of D,RD st F is_left_distributive_wrt G holds
F /\/ RD is_left_distributive_wrt G /\/ RD

proof end;

theorem Th10: :: FILTER_1:10

for D being non empty set
for RD being Equivalence_Relation of D
for F, G being BinOp of D,RD st F is_right_distributive_wrt G holds
F /\/ RD is_right_distributive_wrt G /\/ RD

proof end;

theorem :: FILTER_1:11

for D being non empty set
for RD being Equivalence_Relation of D
for F, G being BinOp of D,RD st F is_distributive_wrt G holds
F /\/ RD is_distributive_wrt G /\/ RD

proof end;

theorem Th12: :: FILTER_1:12

for D being non empty set
for RD being Equivalence_Relation of D
for F, G being BinOp of D,RD st F absorbs G holds
F /\/ RD absorbs G /\/ RD

proof end;

theorem Th13: :: FILTER_1:13

for I being I_Lattice
for FI being Filter of I holds the L_join of I is BinOp of the carrier of I, equivalence_wrt FI

proof end;

theorem Th14: :: FILTER_1:14

for I being I_Lattice
for FI being Filter of I holds the L_meet of I is BinOp of the carrier of I, equivalence_wrt FI

proof end;

definition

let L be Lattice;
let F be Filter of L;
assume A1: L is I_Lattice ;
func L /\/ F -> strict Lattice means :Def5: :: FILTER_1:def 5
for R being Equivalence_Relation of the carrier of L st R = equivalence_wrt F holds
it = LattStr(# (Class R),( the L_join of L /\/ R),( the L_meet of L /\/ R) #);
existence

proof end;

proof end;

end;

:: deftheorem Def5 defines /\/ FILTER_1:def 5 :

definition

let L be Lattice;
let F be Filter of L;
let a be Element of L;
assume A1: L is I_Lattice ;
func a /\/ F -> Element of (L /\/ F) means :Def6: :: FILTER_1:def 6
for R being Equivalence_Relation of the carrier of L st R = equivalence_wrt F holds
it = Class (R,a);
existence

proof end;

proof end;

end;

:: deftheorem Def6 defines /\/ FILTER_1:def 6 :

theorem Th15: :: FILTER_1:15

for I being I_Lattice
for FI being Filter of I
for i, j being Element of I holds
( (i /\/ FI) "\/" (j /\/ FI) = (i "\/" j) /\/ FI & (i /\/ FI) "/\" (j /\/ FI) = (i "/\" j) /\/ FI )

proof end;

theorem Th16: :: FILTER_1:16

for I being I_Lattice
for FI being Filter of I
for i, j being Element of I holds
( i /\/ FI [= j /\/ FI iff i => j in FI )

proof end;

theorem Th17: :: FILTER_1:17

for I being I_Lattice
for i, j, k being Element of I holds (i "/\" j) => k = i => (j => k)

proof end;

theorem Th18: :: FILTER_1:18

for I being I_Lattice
for FI being Filter of I st I is lower-bounded holds
( I /\/ FI is 0_Lattice & Bottom (I /\/ FI) = (Bottom I) /\/ FI )

proof end;

theorem Th19: :: FILTER_1:19

for I being I_Lattice
for FI being Filter of I holds
( I /\/ FI is 1_Lattice & Top (I /\/ FI) = (Top I) /\/ FI )

proof end;

theorem :: FILTER_1:20

canceled;

registration

let I be I_Lattice;
let FI be Filter of I;
cluster I /\/ FI -> strict implicative ;
coherence

proof end;

end;

theorem :: FILTER_1:21

for B being B_Lattice
for FB being Filter of B holds B /\/ FB is B_Lattice

proof end;

definition

let D1, D2 be set ;
let f1 be BinOp of D1;
let f2 be BinOp of D2;
:: original: |:
redefine func |:f1,f2:| -> BinOp of [:D1,D2:];
coherence

proof end;

end;

theorem Th22: :: FILTER_1:22

for D1, D2 being non empty set
for a1, b1 being Element of D1
for a2, b2 being Element of D2
for f1 being BinOp of D1
for f2 being BinOp of D2 holds |:f1,f2:| . ([a1,a2],[b1,b2]) = [(f1 . (a1,b1)),(f2 . (a2,b2))]

proof end;

scheme :: FILTER_1:sch 4
AuxCart1{ F₁() -> non empty set , F₂() -> non empty set , P₁[ set ] } :

for d being Element of [:F₁(),F₂():] holds P₁[d]

provided

A1: for d1 being Element of F₁()
for d2 being Element of F₂() holds P₁[[d1,d2]]

proof end;

scheme :: FILTER_1:sch 5
AuxCart2{ F₁() -> non empty set , F₂() -> non empty set , P₁[ set , set ] } :

for d, d9 being Element of [:F₁(),F₂():] holds P₁[d,d9]

provided

A1: for d1, d19 being Element of F₁()
for d2, d29 being Element of F₂() holds P₁[[d1,d2],[d19,d29]]

proof end;

scheme :: FILTER_1:sch 6
AuxCart3{ F₁() -> non empty set , F₂() -> non empty set , P₁[ set , set , set ] } :

for a, b, c being Element of [:F₁(),F₂():] holds P₁[a,b,c]

provided

A1: for a1, b1, c1 being Element of F₁()
for a2, b2, c2 being Element of F₂() holds P₁[[a1,a2],[b1,b2],[c1,c2]]

proof end;

theorem Th23: :: FILTER_1:23

for D1, D2 being non empty set
for f1 being BinOp of D1
for f2 being BinOp of D2 holds
( ( f1 is commutative & f2 is commutative ) iff |:f1,f2:| is commutative )

proof end;

theorem Th24: :: FILTER_1:24

for D1, D2 being non empty set
for f1 being BinOp of D1
for f2 being BinOp of D2 holds
( ( f1 is associative & f2 is associative ) iff |:f1,f2:| is associative )

proof end;

theorem Th25: :: FILTER_1:25

for D1, D2 being non empty set
for a1 being Element of D1
for a2 being Element of D2
for f1 being BinOp of D1
for f2 being BinOp of D2 holds
( ( a1 is_a_left_unity_wrt f1 & a2 is_a_left_unity_wrt f2 ) iff [a1,a2] is_a_left_unity_wrt |:f1,f2:| )

proof end;

theorem Th26: :: FILTER_1:26

for D1, D2 being non empty set
for a1 being Element of D1
for a2 being Element of D2
for f1 being BinOp of D1
for f2 being BinOp of D2 holds
( ( a1 is_a_right_unity_wrt f1 & a2 is_a_right_unity_wrt f2 ) iff [a1,a2] is_a_right_unity_wrt |:f1,f2:| )

proof end;

theorem :: FILTER_1:27

for D1, D2 being non empty set
for a1 being Element of D1
for a2 being Element of D2
for f1 being BinOp of D1
for f2 being BinOp of D2 holds
( ( a1 is_a_unity_wrt f1 & a2 is_a_unity_wrt f2 ) iff [a1,a2] is_a_unity_wrt |:f1,f2:| )

proof end;

theorem Th28: :: FILTER_1:28

for D1, D2 being non empty set
for f1, g1 being BinOp of D1
for f2, g2 being BinOp of D2 holds
( ( f1 is_left_distributive_wrt g1 & f2 is_left_distributive_wrt g2 ) iff |:f1,f2:| is_left_distributive_wrt |:g1,g2:| )

proof end;

theorem Th29: :: FILTER_1:29

for D1, D2 being non empty set
for f1, g1 being BinOp of D1
for f2, g2 being BinOp of D2 holds
( ( f1 is_right_distributive_wrt g1 & f2 is_right_distributive_wrt g2 ) iff |:f1,f2:| is_right_distributive_wrt |:g1,g2:| )

proof end;

theorem Th30: :: FILTER_1:30

for D1, D2 being non empty set
for f1, g1 being BinOp of D1
for f2, g2 being BinOp of D2 holds
( ( f1 is_distributive_wrt g1 & f2 is_distributive_wrt g2 ) iff |:f1,f2:| is_distributive_wrt |:g1,g2:| )

proof end;

theorem Th31: :: FILTER_1:31

for D1, D2 being non empty set
for f1, g1 being BinOp of D1
for f2, g2 being BinOp of D2 holds
( ( f1 absorbs g1 & f2 absorbs g2 ) iff |:f1,f2:| absorbs |:g1,g2:| )

proof end;

definition

let L1, L2 be non empty LattStr ;
func [:L1,L2:] -> strict LattStr equals :: FILTER_1:def 7
LattStr(# [: the carrier of L1, the carrier of L2:],|: the L_join of L1, the L_join of L2:|,|: the L_meet of L1, the L_meet of L2:| #);
correctness ;

end;

:: deftheorem defines [: FILTER_1:def 7 :

registration

let L1, L2 be non empty LattStr ;
cluster [:L1,L2:] -> non empty strict ;
coherence ;

end;

definition

let L be Lattice;
func LattRel L -> Relation equals :: FILTER_1:def 8
{ [p,q] where p, q is Element of L : p [= q } ;
coherence

proof end;

end;

:: deftheorem defines LattRel FILTER_1:def 8 :

theorem Th32: :: FILTER_1:32

for L being Lattice
for p, q being Element of L holds
( [p,q] in LattRel L iff p [= q )

proof end;

theorem Th33: :: FILTER_1:33

for L being Lattice holds
( dom (LattRel L) = the carrier of L & rng (LattRel L) = the carrier of L & field (LattRel L) = the carrier of L )

proof end;

definition

let L1, L2 be Lattice;
pred L1,L2 are_isomorphic means :: FILTER_1:def 9
LattRel L1, LattRel L2 are_isomorphic ;
reflexivity by WELLORD1:48;
symmetry by WELLORD1:50;

end;

:: deftheorem defines are_isomorphic FILTER_1:def 9 :

registration

let L1, L2 be Lattice;
cluster [:L1,L2:] -> strict Lattice-like ;
coherence

proof end;

end;

theorem :: FILTER_1:34

for L1, L2, L3 being Lattice st L1,L2 are_isomorphic & L2,L3 are_isomorphic holds
L1,L3 are_isomorphic

proof end;

theorem :: FILTER_1:35

for L1, L2 being non empty LattStr st [:L1,L2:] is Lattice holds
( L1 is Lattice & L2 is Lattice )

proof end;

definition

let L1, L2 be Lattice;
let a be Element of L1;
let b be Element of L2;
:: original: [
redefine func [a,b] -> Element of [:L1,L2:];
coherence

proof end;

end;

theorem :: FILTER_1:36

for L1, L2 being Lattice
for p1, q1 being Element of L1
for p2, q2 being Element of L2 holds
( [p1,p2] "\/" [q1,q2] = [(p1 "\/" q1),(p2 "\/" q2)] & [p1,p2] "/\" [q1,q2] = [(p1 "/\" q1),(p2 "/\" q2)] ) by Th22;

theorem Th37: :: FILTER_1:37

for L1, L2 being Lattice
for p1, q1 being Element of L1
for p2, q2 being Element of L2 holds
( [p1,p2] [= [q1,q2] iff ( p1 [= q1 & p2 [= q2 ) )

proof end;

theorem :: FILTER_1:38

for L1, L2 being Lattice holds
( ( L1 is modular & L2 is modular ) iff [:L1,L2:] is modular )

proof end;

theorem Th39: :: FILTER_1:39

for L1, L2 being Lattice holds
( ( L1 is D_Lattice & L2 is D_Lattice ) iff [:L1,L2:] is D_Lattice )

proof end;

theorem Th40: :: FILTER_1:40

for L1, L2 being Lattice holds
( ( L1 is lower-bounded & L2 is lower-bounded ) iff [:L1,L2:] is lower-bounded )

proof end;

theorem Th41: :: FILTER_1:41

for L1, L2 being Lattice holds
( ( L1 is upper-bounded & L2 is upper-bounded ) iff [:L1,L2:] is upper-bounded )

proof end;

theorem Th42: :: FILTER_1:42

for L1, L2 being Lattice holds
( ( L1 is bounded & L2 is bounded ) iff [:L1,L2:] is bounded )

proof end;

theorem Th43: :: FILTER_1:43

for L1, L2 being Lattice st L1 is 0_Lattice & L2 is 0_Lattice holds
Bottom [:L1,L2:] = [(Bottom L1),(Bottom L2)]

proof end;

theorem Th44: :: FILTER_1:44

for L1, L2 being Lattice st L1 is 1_Lattice & L2 is 1_Lattice holds
Top [:L1,L2:] = [(Top L1),(Top L2)]

proof end;

theorem Th45: :: FILTER_1:45

for L1, L2 being Lattice
for p1, q1 being Element of L1
for p2, q2 being Element of L2 st L1 is 01_Lattice & L2 is 01_Lattice holds
( ( p1 is_a_complement_of q1 & p2 is_a_complement_of q2 ) iff [p1,p2] is_a_complement_of [q1,q2] )

proof end;

theorem Th46: :: FILTER_1:46

for L1, L2 being Lattice holds
( ( L1 is C_Lattice & L2 is C_Lattice ) iff [:L1,L2:] is C_Lattice )

proof end;

theorem :: FILTER_1:47

for L1, L2 being Lattice holds
( ( L1 is B_Lattice & L2 is B_Lattice ) iff [:L1,L2:] is B_Lattice )

proof end;

theorem :: FILTER_1:48

for L1, L2 being Lattice holds
( ( L1 is implicative & L2 is implicative ) iff [:L1,L2:] is implicative )

proof end;

theorem :: FILTER_1:49

for L1, L2 being Lattice holds [:L1,L2:] .: = [:(L1 .:),(L2 .:):] ;

theorem :: FILTER_1:50

for L1, L2 being Lattice holds [:L1,L2:],[:L2,L1:] are_isomorphic

proof end;

theorem Th51: :: FILTER_1:51

for B being B_Lattice
for a, b being Element of B holds a <=> b = (a "/\" b) "\/" ((a `) "/\" (b `))

proof end;

theorem Th52: :: FILTER_1:52

for B being B_Lattice
for a, b being Element of B holds
( (a => b) ` = a "/\" (b `) & (a <=> b) ` = (a "/\" (b `)) "\/" ((a `) "/\" b) & (a <=> b) ` = a <=> (b `) & (a <=> b) ` = (a `) <=> b )

proof end;

theorem Th53: :: FILTER_1:53

for B being B_Lattice
for a, b, c being Element of B st a <=> b = a <=> c holds
b = c

proof end;

theorem Th54: :: FILTER_1:54

for B being B_Lattice
for a, b being Element of B holds a <=> (a <=> b) = b

proof end;

theorem :: FILTER_1:55

for I being I_Lattice
for i, j being Element of I holds
( (i "\/" j) => i = j => i & i => (i "/\" j) = i => j )

proof end;

theorem Th56: :: FILTER_1:56

for I being I_Lattice
for i, j, k being Element of I holds
( i => j [= i => (j "\/" k) & i => j [= (i "/\" k) => j & i => j [= i => (k "\/" j) & i => j [= (k "/\" i) => j )

proof end;

Lm1: for I being I_Lattice
for FI being Filter of I
for i, j, k being Element of I st i => j in FI holds
( i => (j "\/" k) in FI & i => (k "\/" j) in FI & (i "/\" k) => j in FI & (k "/\" i) => j in FI )

proof end;

theorem Th57: :: FILTER_1:57

for I being I_Lattice
for i, k, j being Element of I holds (i => k) "/\" (j => k) [= (i "\/" j) => k

proof end;

Lm2: for I being I_Lattice
for FI being Filter of I
for i, k, j being Element of I st i => k in FI & j => k in FI holds
(i "\/" j) => k in FI

proof end;

theorem Th58: :: FILTER_1:58

for I being I_Lattice
for i, j, k being Element of I holds (i => j) "/\" (i => k) [= i => (j "/\" k)

proof end;

Lm3: for I being I_Lattice
for FI being Filter of I
for i, j, k being Element of I st i => j in FI & i => k in FI holds
i => (j "/\" k) in FI

proof end;

theorem Th59: :: FILTER_1:59

for I being I_Lattice
for FI being Filter of I
for i1, i2, j1, j2 being Element of I st i1 <=> i2 in FI & j1 <=> j2 in FI holds
( (i1 "\/" j1) <=> (i2 "\/" j2) in FI & (i1 "/\" j1) <=> (i2 "/\" j2) in FI )

proof end;

Lm4: for I being I_Lattice
for FI being Filter of I
for i, j being Element of I holds
( i in Class ((equivalence_wrt FI),j) iff i <=> j in FI )

proof end;

theorem Th60: :: FILTER_1:60

for I being I_Lattice
for FI being Filter of I
for i, k, j being Element of I st i in Class ((equivalence_wrt FI),k) & j in Class ((equivalence_wrt FI),k) holds
( i "\/" j in Class ((equivalence_wrt FI),k) & i "/\" j in Class ((equivalence_wrt FI),k) )

proof end;

theorem Th61: :: FILTER_1:61

for B being B_Lattice
for c, d being Element of B holds
( c "\/" (c <=> d) in Class ((equivalence_wrt <.d.)),c) & ( for b being Element of B st b in Class ((equivalence_wrt <.d.)),c) holds
b [= c "\/" (c <=> d) ) )

proof end;

theorem :: FILTER_1:62

for B being B_Lattice
for a being Element of B holds B,[:(B /\/ <.a.)),(latt <.a.)):] are_isomorphic

proof end;