:: Finite Join and Finite Meet, and Dual Lattices
:: by Andrzej Trybulec
::
:: Received August 10, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

theorem Th1: :: LATTICE2:1

for A being set
for C being non empty set
for B being Subset of A
for g being Function of A,C holds dom (g | B) = B

proof end;

theorem Th2: :: LATTICE2:2

for A being set
for C being non empty set
for B being Subset of A
for f, g being Function of A,C holds
( f | B = g | B iff for x being Element of A st x in B holds
g . x = f . x )

proof end;

theorem Th3: :: LATTICE2:3

for A being set
for C being non empty set
for f, g being Function of A,C
for B being set holds f +* (g | B) is Function of A,C

proof end;

theorem Th4: :: LATTICE2:4

for A being set
for C being non empty set
for B being Subset of A
for f, g being Function of A,C holds (g | B) +* f = f

proof end;

theorem Th5: :: LATTICE2:5

for f, g being Function st g c= f holds
f +* g = f

proof end;

theorem :: LATTICE2:6

for A being set
for C being non empty set
for B being Subset of A
for f being Function of A,C holds f +* (f | B) = f by Th5, RELAT_1:59;

theorem Th7: :: LATTICE2:7

for A being set
for C being non empty set
for B being Subset of A
for f, g being Function of A,C st ( for x being Element of A st x in B holds
g . x = f . x ) holds
f +* (g | B) = f

proof end;

theorem :: LATTICE2:8

for A being set
for C being non empty set
for f, g being Function of A,C
for B being Element of Fin A holds (g | B) +* f = f

proof end;

theorem Th9: :: LATTICE2:9

for A being set
for C being non empty set
for g being Function of A,C
for B being Element of Fin A holds dom (g | B) = B

proof end;

theorem Th10: :: LATTICE2:10

for A being set
for C being non empty set
for f, g being Function of A,C
for B being Element of Fin A st ( for x being Element of A st x in B holds
g . x = f . x ) holds
f +* (g | B) = f

proof end;

definition

let D be non empty set ;
let o, o9 be BinOp of D;

pred o absorbs o9 means :: LATTICE2:def 1
for x, y being Element of D holds o . (x,(o9 . (x,y))) = x;

end;

:: deftheorem defines absorbs LATTICE2:def 1 :
for D being non empty set
for o, o9 being BinOp of D holds
( o absorbs o9 iff for x, y being Element of D holds o . (x,(o9 . (x,y))) = x );

notation

let D be non empty set ;
let o, o9 be BinOp of D;
antonym o doesn't_absorb o9 for o absorbs o9;

end;

theorem Th11: :: LATTICE2:11

for L being non empty LattStr st the L_join of L is commutative & the L_join of L is associative & the L_meet of L is commutative & the L_meet of L is associative & the L_join of L absorbs the L_meet of L & the L_meet of L absorbs the L_join of L holds
L is Lattice-like

proof end;

definition

let L be LattStr ;

func L .: -> strict LattStr equals :: LATTICE2:def 2
LattStr(# the carrier of L, the L_meet of L, the L_join of L #);

correctness
coherence
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is strict LattStr ;
;

end;

:: deftheorem defines .: LATTICE2:def 2 :
for L being LattStr holds L .: = LattStr(# the carrier of L, the L_meet of L, the L_join of L #);

registration

let L be non empty LattStr ;

cluster L .: -> non empty strict ;

coherence
not L .: is empty ;

end;

theorem :: LATTICE2:12

for L being non empty LattStr holds
( the carrier of L = the carrier of (L .:) & the L_join of L = the L_meet of (L .:) & the L_meet of L = the L_join of (L .:) ) ;

theorem :: LATTICE2:13

for L being non empty strict LattStr holds (L .:) .: = L ;

theorem Th14: :: LATTICE2:14

for L being Lattice
for u being Element of L st ( for v being Element of L holds u "\/" v = v ) holds
u = Bottom L

proof end;

theorem Th15: :: LATTICE2:15

for L being Lattice
for u being Element of L st ( for v being Element of L holds the L_join of L . (u,v) = v ) holds
u = Bottom L

proof end;

theorem Th16: :: LATTICE2:16

for L being Lattice
for u being Element of L st ( for v being Element of L holds u "/\" v = v ) holds
u = Top L

proof end;

theorem Th17: :: LATTICE2:17

for L being Lattice
for u being Element of L st ( for v being Element of L holds the L_meet of L . (u,v) = v ) holds
u = Top L

proof end;

registration

let L be Lattice;

cluster the L_join of L -> idempotent ;

coherence
the L_join of L is idempotent

proof end;

end;

registration

let L be non empty join-commutative \/-SemiLattStr ;

cluster the L_join of L -> commutative ;

coherence
the L_join of L is commutative

proof end;

end;

theorem Th18: :: LATTICE2:18

for L being Lattice st the L_join of L is having_a_unity holds
Bottom L = the_unity_wrt the L_join of L

proof end;

registration

let L be non empty join-associative \/-SemiLattStr ;

cluster the L_join of L -> associative ;

coherence
the L_join of L is associative

proof end;

end;

registration

let L be Lattice;

cluster the L_meet of L -> idempotent ;

coherence
the L_meet of L is idempotent

proof end;

end;

registration

let L be non empty meet-commutative /\-SemiLattStr ;

cluster the L_meet of L -> commutative ;

coherence
the L_meet of L is commutative

proof end;

end;

registration

let L be non empty meet-associative /\-SemiLattStr ;

cluster the L_meet of L -> associative ;

coherence
the L_meet of L is associative

proof end;

end;

theorem Th19: :: LATTICE2:19

for L being Lattice st the L_meet of L is having_a_unity holds
Top L = the_unity_wrt the L_meet of L

proof end;

theorem Th20: :: LATTICE2:20

for L being Lattice holds the L_join of L is_distributive_wrt the L_join of L

proof end;

theorem :: LATTICE2:21

for L being Lattice st L is D_Lattice holds
the L_join of L is_distributive_wrt the L_meet of L

proof end;

theorem Th22: :: LATTICE2:22

for L being Lattice st the L_join of L is_distributive_wrt the L_meet of L holds
L is distributive

proof end;

theorem Th23: :: LATTICE2:23

for L being Lattice st L is D_Lattice holds
the L_meet of L is_distributive_wrt the L_join of L

proof end;

theorem :: LATTICE2:24

for L being Lattice st the L_meet of L is_distributive_wrt the L_join of L holds
L is distributive by BINOP_1:12;

theorem Th25: :: LATTICE2:25

for L being Lattice holds the L_meet of L is_distributive_wrt the L_meet of L

proof end;

theorem Th26: :: LATTICE2:26

for L being Lattice holds the L_join of L absorbs the L_meet of L

proof end;

theorem Th27: :: LATTICE2:27

for L being Lattice holds the L_meet of L absorbs the L_join of L

proof end;

definition

let A be non empty set ;
let L be Lattice;
let B be Element of Fin A;
let f be Function of A, the carrier of L;

func FinJoin (B,f) -> Element of L equals :: LATTICE2:def 3
the L_join of L $$ (B,f);

correctness
coherence
the L_join of L $$ (B,f) is Element of L;
;

func FinMeet (B,f) -> Element of L equals :: LATTICE2:def 4
the L_meet of L $$ (B,f);

correctness
coherence
the L_meet of L $$ (B,f) is Element of L;
;

end;

:: deftheorem defines FinJoin LATTICE2:def 3 :
for A being non empty set
for L being Lattice
for B being Element of Fin A
for f being Function of A, the carrier of L holds FinJoin (B,f) = the L_join of L $$ (B,f);

:: deftheorem defines FinMeet LATTICE2:def 4 :
for A being non empty set
for L being Lattice
for B being Element of Fin A
for f being Function of A, the carrier of L holds FinMeet (B,f) = the L_meet of L $$ (B,f);

theorem Th28: :: LATTICE2:28

for L being Lattice
for A being non empty set
for x being Element of A
for B being Element of Fin A
for f being Function of A, the carrier of L st x in B holds
f . x [= FinJoin (B,f) by SETWISEO:22;

theorem Th29: :: LATTICE2:29

for L being Lattice
for u being Element of L
for A being non empty set
for B being Element of Fin A
for f being Function of A, the carrier of L st ex x being Element of A st
( x in B & u [= f . x ) holds
u [= FinJoin (B,f)

proof end;

theorem :: LATTICE2:30

for L being Lattice
for u being Element of L
for A being non empty set
for B being Element of Fin A
for f being Function of A, the carrier of L st ( for x being Element of A st x in B holds
f . x = u ) & B <> {} holds
FinJoin (B,f) = u by SETWISEO:24;

theorem :: LATTICE2:31

for L being Lattice
for u being Element of L
for A being non empty set
for B being Element of Fin A
for f being Function of A, the carrier of L st FinJoin (B,f) [= u holds
for x being Element of A st x in B holds
f . x [= u

proof end;

theorem Th32: :: LATTICE2:32

for L being Lattice
for u being Element of L
for A being non empty set
for B being Element of Fin A
for f being Function of A, the carrier of L st B <> {} & ( for x being Element of A st x in B holds
f . x [= u ) holds
FinJoin (B,f) [= u

proof end;

theorem :: LATTICE2:33

for L being Lattice
for A being non empty set
for B being Element of Fin A
for f, g being Function of A, the carrier of L st B <> {} & ( for x being Element of A st x in B holds
f . x [= g . x ) holds
FinJoin (B,f) [= FinJoin (B,g)

proof end;

theorem Th34: :: LATTICE2:34

for L being Lattice
for A being non empty set
for B being Element of Fin A
for f, g being Function of A, the carrier of L st B <> {} & f | B = g | B holds
FinJoin (B,f) = FinJoin (B,g)

proof end;

theorem :: LATTICE2:35

for L being Lattice
for v being Element of L
for A being non empty set
for B being Element of Fin A
for f being Function of A, the carrier of L st B <> {} holds
v "\/" (FinJoin (B,f)) = FinJoin (B,( the L_join of L [;] (v,f))) by Th20, SETWISEO:27;

registration

let L be Lattice;

cluster L .: -> strict Lattice-like ;

coherence
L .: is Lattice-like

proof end;

end;

theorem :: LATTICE2:36

for A being non empty set
for L being Lattice
for B being Element of Fin A
for f being Function of A, the carrier of L
for f9 being Function of A, the carrier of (L .:) st f = f9 holds
( FinJoin (B,f) = FinMeet (B,f9) & FinMeet (B,f) = FinJoin (B,f9) ) ;

theorem Th37: :: LATTICE2:37

for L being Lattice
for a, b being Element of L
for a9, b9 being Element of (L .:) st a = a9 & b = b9 holds
( a "/\" b = a9 "\/" b9 & a "\/" b = a9 "/\" b9 ) ;

theorem Th38: :: LATTICE2:38

for L being Lattice
for a, b being Element of L st a [= b holds
for a9, b9 being Element of (L .:) st a = a9 & b = b9 holds
b9 [= a9

proof end;

theorem Th39: :: LATTICE2:39

for L being Lattice
for a, b being Element of L
for a9, b9 being Element of (L .:) st a9 [= b9 & a = a9 & b = b9 holds
b [= a

proof end;

:: Dualizations

theorem Th40: :: LATTICE2:40

for L being Lattice
for A being non empty set
for x being Element of A
for B being Element of Fin A
for f being Function of A, the carrier of L st x in B holds
FinMeet (B,f) [= f . x

proof end;

theorem Th41: :: LATTICE2:41

for L being Lattice
for u being Element of L
for A being non empty set
for B being Element of Fin A
for f being Function of A, the carrier of L st ex x being Element of A st
( x in B & f . x [= u ) holds
FinMeet (B,f) [= u

proof end;

theorem :: LATTICE2:42

for L being Lattice
for u being Element of L
for A being non empty set
for B being Element of Fin A
for f being Function of A, the carrier of L st ( for x being Element of A st x in B holds
f . x = u ) & B <> {} holds
FinMeet (B,f) = u by SETWISEO:24;

theorem :: LATTICE2:43

for L being Lattice
for v being Element of L
for A being non empty set
for B being Element of Fin A
for f being Function of A, the carrier of L st B <> {} holds
v "/\" (FinMeet (B,f)) = FinMeet (B,( the L_meet of L [;] (v,f))) by Th25, SETWISEO:27;

theorem :: LATTICE2:44

for L being Lattice
for u being Element of L
for A being non empty set
for B being Element of Fin A
for f being Function of A, the carrier of L st u [= FinMeet (B,f) holds
for x being Element of A st x in B holds
u [= f . x

proof end;

theorem :: LATTICE2:45

for L being Lattice
for A being non empty set
for B being Element of Fin A
for f, g being Function of A, the carrier of L st B <> {} & f | B = g | B holds
FinMeet (B,f) = FinMeet (B,g)

proof end;

theorem Th46: :: LATTICE2:46

for L being Lattice
for u being Element of L
for A being non empty set
for B being Element of Fin A
for f being Function of A, the carrier of L st B <> {} & ( for x being Element of A st x in B holds
u [= f . x ) holds
u [= FinMeet (B,f)

proof end;

theorem :: LATTICE2:47

for L being Lattice
for A being non empty set
for B being Element of Fin A
for f, g being Function of A, the carrier of L st B <> {} & ( for x being Element of A st x in B holds
f . x [= g . x ) holds
FinMeet (B,f) [= FinMeet (B,g)

proof end;

theorem :: LATTICE2:48

for L being Lattice holds
( L is lower-bounded iff L .: is upper-bounded )

proof end;

theorem Th49: :: LATTICE2:49

for L being Lattice holds
( L is upper-bounded iff L .: is lower-bounded )

proof end;

theorem :: LATTICE2:50

for L being Lattice holds
( L is D_Lattice iff L .: is D_Lattice ) by Th22, Th23;

theorem Th51: :: LATTICE2:51

for L being 0_Lattice holds Bottom L is_a_unity_wrt the L_join of L

proof end;

registration

let L be 0_Lattice;

cluster the L_join of L -> having_a_unity ;

coherence
the L_join of L is having_a_unity

proof end;

end;

theorem :: LATTICE2:52

for L being 0_Lattice holds Bottom L = the_unity_wrt the L_join of L by Th18;

theorem Th53: :: LATTICE2:53

for A being non empty set
for B being Element of Fin A
for L being 0_Lattice
for f, g being Function of A, the carrier of L st f | B = g | B holds
FinJoin (B,f) = FinJoin (B,g)

proof end;

theorem Th54: :: LATTICE2:54

for A being non empty set
for B being Element of Fin A
for L being 0_Lattice
for f being Function of A, the carrier of L
for u being Element of L st ( for x being Element of A st x in B holds
f . x [= u ) holds
FinJoin (B,f) [= u

proof end;

theorem :: LATTICE2:55

for A being non empty set
for B being Element of Fin A
for L being 0_Lattice
for f, g being Function of A, the carrier of L st ( for x being Element of A st x in B holds
f . x [= g . x ) holds
FinJoin (B,f) [= FinJoin (B,g)

proof end;

theorem Th56: :: LATTICE2:56

for L being 1_Lattice holds Top L is_a_unity_wrt the L_meet of L

proof end;

registration

let L be 1_Lattice;

cluster the L_meet of L -> having_a_unity ;

coherence
the L_meet of L is having_a_unity

proof end;

end;

theorem :: LATTICE2:57

for L being 1_Lattice holds Top L = the_unity_wrt the L_meet of L by Th19;

theorem :: LATTICE2:58

for A being non empty set
for B being Element of Fin A
for L being 1_Lattice
for f, g being Function of A, the carrier of L st f | B = g | B holds
FinMeet (B,f) = FinMeet (B,g)

proof end;

theorem Th59: :: LATTICE2:59

for A being non empty set
for B being Element of Fin A
for L being 1_Lattice
for f being Function of A, the carrier of L
for u being Element of L st ( for x being Element of A st x in B holds
u [= f . x ) holds
u [= FinMeet (B,f)

proof end;

theorem :: LATTICE2:60

for A being non empty set
for B being Element of Fin A
for L being 1_Lattice
for f, g being Function of A, the carrier of L st ( for x being Element of A st x in B holds
f . x [= g . x ) holds
FinMeet (B,f) [= FinMeet (B,g)

proof end;

theorem :: LATTICE2:61

for L being 0_Lattice holds Bottom L = Top (L .:)

proof end;

theorem :: LATTICE2:62

for L being 1_Lattice holds Top L = Bottom (L .:)

proof end;

::
:: Distributive lattices with the minimal element
::

definition

mode D0_Lattice is distributive lower-bounded Lattice;

end;

theorem :: LATTICE2:63

for L being D0_Lattice holds the L_meet of L is_distributive_wrt the L_join of L by Th23;

theorem Th64: :: LATTICE2:64

for A being non empty set
for B being Element of Fin A
for L being D0_Lattice
for f being Function of A, the carrier of L
for u being Element of L holds the L_meet of L . (u,(FinJoin (B,f))) = FinJoin (B,( the L_meet of L [;] (u,f)))

proof end;

theorem :: LATTICE2:65

for A being non empty set
for B being Element of Fin A
for L being D0_Lattice
for f, g being Function of A, the carrier of L
for u being Element of L st ( for x being Element of A st x in B holds
g . x = u "/\" (f . x) ) holds
u "/\" (FinJoin (B,f)) = FinJoin (B,g)

proof end;

theorem :: LATTICE2:66

for A being non empty set
for B being Element of Fin A
for L being D0_Lattice
for f being Function of A, the carrier of L
for u being Element of L holds u "/\" (FinJoin (B,f)) = FinJoin (B,( the L_meet of L [;] (u,f))) by Th64;

:: Heyting Lattices

definition

let IT be Lattice;

attr IT is Heyting means :: LATTICE2:def 5
( IT is implicative & IT is lower-bounded );

end;

:: deftheorem defines Heyting LATTICE2:def 5 :
for IT being Lattice holds
( IT is Heyting iff ( IT is implicative & IT is lower-bounded ) );

registration

cluster non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like Heyting for LattStr ;

existence
ex b₁ being Lattice st b₁ is Heyting

proof end;

end;

registration

cluster non empty Lattice-like Heyting -> lower-bounded implicative for LattStr ;

coherence
for b₁ being Lattice st b₁ is Heyting holds
( b₁ is implicative & b₁ is lower-bounded ) ;

cluster non empty Lattice-like lower-bounded implicative -> Heyting for LattStr ;

coherence
for b₁ being Lattice st b₁ is implicative & b₁ is lower-bounded holds
b₁ is Heyting ;

end;

definition

mode H_Lattice is Heyting Lattice;

end;

registration

cluster non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like Heyting for LattStr ;

existence
ex b₁ being Lattice st
( b₁ is Heyting & b₁ is strict )

proof end;

end;

theorem :: LATTICE2:67

for L being 0_Lattice holds
( L is H_Lattice iff for x, z being Element of L ex y being Element of L st
( x "/\" y [= z & ( for v being Element of L st x "/\" v [= z holds
v [= y ) ) )

proof end;

theorem :: LATTICE2:68

for L being Lattice holds
( L is finite iff L .: is finite ) ;

registration

cluster non empty finite Lattice-like -> lower-bounded for LattStr ;

coherence
for b₁ being Lattice st b₁ is finite holds
b₁ is lower-bounded

proof end;

cluster non empty finite Lattice-like -> upper-bounded for LattStr ;

coherence
for b₁ being Lattice st b₁ is finite holds
b₁ is upper-bounded

proof end;

end;

registration

cluster non empty finite Lattice-like -> bounded for LattStr ;

coherence
for b₁ being Lattice st b₁ is finite holds
b₁ is bounded ;

end;

registration

cluster non empty finite Lattice-like distributive -> Heyting for LattStr ;

coherence
for b₁ being Lattice st b₁ is distributive & b₁ is finite holds
b₁ is Heyting

proof end;

end;