:: by Piotr Rudnicki
::
:: Copyright (c) 1998-2019 Association of Mizar Users

Lm1: for L being complete LATTICE
for X being set st X c= bool the carrier of L holds
"/\" ((),L) = "/\" ( { (inf Y) where Y is Subset of L : Y in X } ,L)

proof end;

Lm2: for L being complete LATTICE
for X being set st X c= bool the carrier of L holds
"\/" ((),L) = "\/" ( { (sup Y) where Y is Subset of L : Y in X } ,L)

proof end;

theorem Th1: :: WAYBEL22:1
for L being upper-bounded Semilattice
for F being non empty directed Subset of (InclPoset (Filt L)) holds sup F = union F
proof end;

theorem Th2: :: WAYBEL22:2
for L, S, T being non empty complete Poset
for f being CLHomomorphism of L,S
for g being CLHomomorphism of S,T holds g * f is CLHomomorphism of L,T
proof end;

theorem Th3: :: WAYBEL22:3
for L being non empty RelStr holds id L is infs-preserving
proof end;

theorem Th4: :: WAYBEL22:4
for L being non empty RelStr holds id L is directed-sups-preserving
proof end;

theorem Th5: :: WAYBEL22:5
for L being non empty complete Poset holds id L is CLHomomorphism of L,L
proof end;

theorem Th6: :: WAYBEL22:6
for L being non empty upper-bounded with_infima Poset holds InclPoset (Filt L) is CLSubFrame of BoolePoset the carrier of L
proof end;

registration
let L be non empty upper-bounded with_infima Poset;
coherence
InclPoset (Filt L) is continuous
proof end;
end;

registration
let L be non empty upper-bounded Poset;
cluster -> non empty for Element of the carrier of (InclPoset (Filt L));
coherence
for b1 being Element of (InclPoset (Filt L)) holds not b1 is empty
proof end;
end;

definition
let S be non empty complete continuous Poset;
let A be set ;
pred A is_FreeGen_set_of S means :: WAYBEL22:def 1
for T being non empty complete continuous Poset
for f being Function of A, the carrier of T ex h being CLHomomorphism of S,T st
( h | A = f & ( for h9 being CLHomomorphism of S,T st h9 | A = f holds
h9 = h ) );
end;

:: deftheorem defines is_FreeGen_set_of WAYBEL22:def 1 :
for S being non empty complete continuous Poset
for A being set holds
( A is_FreeGen_set_of S iff for T being non empty complete continuous Poset
for f being Function of A, the carrier of T ex h being CLHomomorphism of S,T st
( h | A = f & ( for h9 being CLHomomorphism of S,T st h9 | A = f holds
h9 = h ) ) );

theorem Th7: :: WAYBEL22:7
for S being non empty complete continuous Poset
for A being set st A is_FreeGen_set_of S holds
A is Subset of S
proof end;

theorem Th8: :: WAYBEL22:8
for S being non empty complete continuous Poset
for A being set st A is_FreeGen_set_of S holds
for h9 being CLHomomorphism of S,S st h9 | A = id A holds
h9 = id S
proof end;

definition
let X be set ;
func FixedUltraFilters X -> Subset-Family of () equals :: WAYBEL22:def 2
{ () where x is Element of () : ex z being Element of X st x = {z} } ;
coherence
{ () where x is Element of () : ex z being Element of X st x = {z} } is Subset-Family of ()
proof end;
end;

:: deftheorem defines FixedUltraFilters WAYBEL22:def 2 :
for X being set holds FixedUltraFilters X = { () where x is Element of () : ex z being Element of X st x = {z} } ;

theorem Th9: :: WAYBEL22:9
for X being set holds FixedUltraFilters X c= Filt ()
proof end;

theorem Th10: :: WAYBEL22:10
for X being set holds card = card X
proof end;

theorem Th11: :: WAYBEL22:11
for X being set
for F being Filter of () holds F = "\/" ( { ("/\" ( { () where x is Element of () : ex z being Element of X st
( x = {z} & z in Y )
}
,(InclPoset (Filt ())))
)
where Y is Subset of X : Y in F
}
,(InclPoset (Filt ())))
proof end;

definition
let X be set ;
let L be non empty complete continuous Poset;
let f be Function of , the carrier of L;
func f -extension_to_hom -> Function of (InclPoset (Filt ())),L means :Def3: :: WAYBEL22:def 3
for Fi being Element of (InclPoset (Filt ())) holds it . Fi = "\/" ( { ("/\" ( { (f . ()) where x is Element of () : ex z being Element of X st
( x = {z} & z in Y )
}
,L)
)
where Y is Subset of X : Y in Fi
}
,L);
existence
ex b1 being Function of (InclPoset (Filt ())),L st
for Fi being Element of (InclPoset (Filt ())) holds b1 . Fi = "\/" ( { ("/\" ( { (f . ()) where x is Element of () : ex z being Element of X st
( x = {z} & z in Y )
}
,L)
)
where Y is Subset of X : Y in Fi
}
,L)
proof end;
uniqueness
for b1, b2 being Function of (InclPoset (Filt ())),L st ( for Fi being Element of (InclPoset (Filt ())) holds b1 . Fi = "\/" ( { ("/\" ( { (f . ()) where x is Element of () : ex z being Element of X st
( x = {z} & z in Y )
}
,L)
)
where Y is Subset of X : Y in Fi
}
,L) ) & ( for Fi being Element of (InclPoset (Filt ())) holds b2 . Fi = "\/" ( { ("/\" ( { (f . ()) where x is Element of () : ex z being Element of X st
( x = {z} & z in Y )
}
,L)
)
where Y is Subset of X : Y in Fi
}
,L) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines -extension_to_hom WAYBEL22:def 3 :
for X being set
for L being non empty complete continuous Poset
for f being Function of , the carrier of L
for b4 being Function of (InclPoset (Filt ())),L holds
( b4 = f -extension_to_hom iff for Fi being Element of (InclPoset (Filt ())) holds b4 . Fi = "\/" ( { ("/\" ( { (f . ()) where x is Element of () : ex z being Element of X st
( x = {z} & z in Y )
}
,L)
)
where Y is Subset of X : Y in Fi
}
,L) );

theorem :: WAYBEL22:12
for X being set
for L being non empty complete continuous Poset
for f being Function of , the carrier of L holds f -extension_to_hom is monotone
proof end;

theorem Th13: :: WAYBEL22:13
for X being set
for L being non empty complete continuous Poset
for f being Function of , the carrier of L holds . (Top (InclPoset (Filt ()))) = Top L
proof end;

registration
let X be set ;
let L be non empty complete continuous Poset;
let f be Function of , the carrier of L;
coherence
proof end;
end;

registration
let X be set ;
let L be non empty complete continuous Poset;
let f be Function of , the carrier of L;
coherence
proof end;
end;

theorem Th14: :: WAYBEL22:14
for X being set
for L being non empty complete continuous Poset
for f being Function of , the carrier of L holds | = f
proof end;

theorem Th15: :: WAYBEL22:15
for X being set
for L being non empty complete continuous Poset
for f being Function of , the carrier of L
for h being CLHomomorphism of InclPoset (Filt ()),L st h | = f holds
h = f -extension_to_hom
proof end;

theorem Th16: :: WAYBEL22:16
for X being set holds FixedUltraFilters X is_FreeGen_set_of InclPoset (Filt ())
proof end;

theorem Th17: :: WAYBEL22:17
for L, M being complete continuous LATTICE
for F, G being set st F is_FreeGen_set_of L & G is_FreeGen_set_of M & card F = card G holds
L,M are_isomorphic
proof end;

:: Representation Theorem for Free Continuous Lattices
theorem :: WAYBEL22:18
for X being set
for L being complete continuous LATTICE
for G being set st G is_FreeGen_set_of L & card G = card X holds
L, InclPoset (Filt ()) are_isomorphic
proof end;