:: LATTICE3 semantic presentation
deffunc H1( LattStr ) -> set = the carrier of $1;
deffunc H2( LattStr ) -> Relation of [:the carrier of $1,the carrier of $1:],the carrier of $1 = the L_join of $1;
deffunc H3( LattStr ) -> Relation of [:the carrier of $1,the carrier of $1:],the carrier of $1 = the L_meet of $1;
definition
let X be
set ;
func BooleLatt X -> strict LattStr means :
Def1:
:: LATTICE3:def 1
( the
carrier of
it = bool X & ( for
Y,
Z being
Subset of
X holds
( the
L_join of
it . Y,
Z = Y \/ Z & the
L_meet of
it . Y,
Z = Y /\ Z ) ) );
existence
ex b1 being strict LattStr st
( the carrier of b1 = bool X & ( for Y, Z being Subset of X holds
( the L_join of b1 . Y,Z = Y \/ Z & the L_meet of b1 . Y,Z = Y /\ Z ) ) )
uniqueness
for b1, b2 being strict LattStr st the carrier of b1 = bool X & ( for Y, Z being Subset of X holds
( the L_join of b1 . Y,Z = Y \/ Z & the L_meet of b1 . Y,Z = Y /\ Z ) ) & the carrier of b2 = bool X & ( for Y, Z being Subset of X holds
( the L_join of b2 . Y,Z = Y \/ Z & the L_meet of b2 . Y,Z = Y /\ Z ) ) holds
b1 = b2
end;
:: deftheorem Def1 defines BooleLatt LATTICE3:def 1 :
theorem Th1: :: LATTICE3:1
theorem Th2: :: LATTICE3:2
theorem Th3: :: LATTICE3:3
theorem Th4: :: LATTICE3:4
theorem :: LATTICE3:5
:: deftheorem defines LattPOSet LATTICE3:def 2 :
theorem Th6: :: LATTICE3:6
:: deftheorem defines % LATTICE3:def 3 :
:: deftheorem defines % LATTICE3:def 4 :
theorem Th7: :: LATTICE3:7
:: deftheorem defines ~ LATTICE3:def 5 :
theorem :: LATTICE3:8
:: deftheorem defines ~ LATTICE3:def 6 :
:: deftheorem defines ~ LATTICE3:def 7 :
theorem Th9: :: LATTICE3:9
:: deftheorem defines is_<=_than LATTICE3:def 8 :
:: deftheorem Def9 defines is_<=_than LATTICE3:def 9 :
:: deftheorem Def10 defines with_suprema LATTICE3:def 10 :
:: deftheorem Def11 defines with_infima LATTICE3:def 11 :
theorem :: LATTICE3:10
theorem :: LATTICE3:11
:: deftheorem Def12 defines complete LATTICE3:def 12 :
theorem Th12: :: LATTICE3:12
:: deftheorem Def13 defines "\/" LATTICE3:def 13 :
:: deftheorem Def14 defines "/\" LATTICE3:def 14 :
theorem Th13: :: LATTICE3:13
theorem Th14: :: LATTICE3:14
theorem Th15: :: LATTICE3:15
theorem Th16: :: LATTICE3:16
theorem Th17: :: LATTICE3:17
theorem Th18: :: LATTICE3:18
theorem Th19: :: LATTICE3:19
:: deftheorem Def15 defines latt LATTICE3:def 15 :
theorem :: LATTICE3:20
:: deftheorem Def16 defines is_less_than LATTICE3:def 16 :
:: deftheorem Def17 defines is_less_than LATTICE3:def 17 :
theorem :: LATTICE3:21
theorem :: LATTICE3:22
:: deftheorem Def18 defines complete LATTICE3:def 18 :
:: deftheorem Def19 defines \/-distributive LATTICE3:def 19 :
:: deftheorem defines /\-distributive LATTICE3:def 20 :
theorem :: LATTICE3:23
theorem Th24: :: LATTICE3:24
theorem Th25: :: LATTICE3:25
theorem Th26: :: LATTICE3:26
theorem Th27: :: LATTICE3:27
theorem Th28: :: LATTICE3:28
theorem :: LATTICE3:29
theorem Th30: :: LATTICE3:30
theorem Th31: :: LATTICE3:31
:: deftheorem Def21 defines "\/" LATTICE3:def 21 :
:: deftheorem defines "/\" LATTICE3:def 22 :
theorem Th32: :: LATTICE3:32
theorem Th33: :: LATTICE3:33
theorem Th34: :: LATTICE3:34
theorem Th35: :: LATTICE3:35
theorem Th36: :: LATTICE3:36
theorem :: LATTICE3:37
theorem Th38: :: LATTICE3:38
theorem :: LATTICE3:39
canceled;
theorem Th40: :: LATTICE3:40
theorem Th41: :: LATTICE3:41
theorem Th42: :: LATTICE3:42
theorem :: LATTICE3:43
theorem :: LATTICE3:44
theorem :: LATTICE3:45
theorem Th46: :: LATTICE3:46
theorem Th47: :: LATTICE3:47
theorem :: LATTICE3:48
theorem :: LATTICE3:49
theorem :: LATTICE3:50
theorem :: LATTICE3:51
theorem Th52: :: LATTICE3:52
theorem :: LATTICE3:53
theorem :: LATTICE3:54
theorem :: LATTICE3:55
Lm3:
now
let D be non
empty set ;
let f be
Function of
bool D,
D;
assume that A1:
for
a being
Element of
D holds
f . {a} = a
and A2:
for
X being
Subset-Family of
D holds
f . (f .: X) = f . (union X)
;
defpred S1[
set ,
set ]
means f . {$1,$2} = $2;
consider R being
Relation of
D such that A3:
for
x,
y being
set holds
(
[x,y] in R iff (
x in D &
y in D &
S1[
x,
y] ) )
from RELSET_1:sch 1();
A4:
dom f = bool D
by FUNCT_2:def 1;
A6:
for
x,
y being
Element of
D for
X being
Subset of
D st
y in X holds
f . (X \/ {x}) = f . { (f . {t,x}) where t is Element of D : t in X }
A12:
R is_reflexive_in D
A14:
R is_antisymmetric_in D
A15:
R is_transitive_in D
proof
let x,
y,
z be
set ;
:: according to RELAT_2:def 8
assume A16:
(
x in D &
y in D &
z in D &
[x,y] in R &
[y,z] in R )
;
then reconsider a =
x,
b =
y,
c =
z as
Element of
D ;
A17:
(
f . {x,y} = y &
f . {y,z} = z )
by A3, A16;
then f . {a,c} =
f . {(f . {a}),(f . {b,c})}
by A1
.=
f . ({a} \/ {b,c})
by A5
.=
f . {a,b,c}
by ENUMSET1:42
.=
f . ({a,b} \/ {c})
by ENUMSET1:43
.=
f . {(f . {a,b}),(f . {c})}
by A5
.=
c
by A1, A17
;
hence
[x,z] in R
by A3;
end;
A18:
dom R = D
by A12, ORDERS_1:98;
field R = D
by A12, ORDERS_1:98;
then reconsider R =
R as
Order of
D by A12, A14, A15, A18, PARTFUN1:def 4, RELAT_2:def 9, RELAT_2:def 12, RELAT_2:def 16;
set A =
RelStr(#
D,
R #);
RelStr(#
D,
R #) is
complete
then reconsider A =
RelStr(#
D,
R #) as non
empty strict complete Poset ;
take L =
latt A;
A25:
(
A is
with_suprema &
A is
with_infima )
by Th12;
then A26:
(
A = LattPOSet L &
LattPOSet L = RelStr(#
H1(
L),
(LattRel L) #) )
by Def15;
hence
H1(
L)
= D
;
let X be
Subset of
L;
reconsider Y =
X as
Subset of
D by A26;
reconsider a =
f . Y as
Element of
(LattPOSet L) by A25, Def15;
set p =
% a;
X is_<=_than a
then A28:
X is_less_than % a
by Th31;
hence
"\/" X = f . X
by A28, Def21;
end;
theorem :: LATTICE3:56