:: WAYBEL17 semantic presentation
:: deftheorem Def1 defines ,... WAYBEL17:def 1 :
theorem Th1: :: WAYBEL17:1
theorem Th2: :: WAYBEL17:2
Lemma69:
for T being up-complete LATTICE
for x being Element of T holds downarrow x is closed_under_directed_sups
Lemma71:
for T being up-complete Scott TopLattice
for x being Element of T holds Cl {x} = downarrow x
Lemma72:
for T being up-complete Scott TopLattice
for x being Element of T holds downarrow x is closed
theorem Th3: :: WAYBEL17:3
theorem Th4: :: WAYBEL17:4
Lemma77:
for S, T being non empty TopSpace-like reflexive transitive antisymmetric up-complete Scott TopRelStr
for f being Function of S,T st f is directed-sups-preserving holds
f is continuous
definition
let S be
up-complete Scott TopLattice,
T be
up-complete Scott TopLattice;
func SCMaps c1,
c2 -> strict full SubRelStr of
MonMaps a1,
a2 means :
Def2:
:: WAYBEL17:def 2
for
f being
Function of
S,
T holds
(
f in the
carrier of
it iff
f is
continuous );
existence
ex b1 being strict full SubRelStr of MonMaps S,T st
for f being Function of S,T holds
( f in the carrier of b1 iff f is continuous )
uniqueness
for b1, b2 being strict full SubRelStr of MonMaps S,T st ( for f being Function of S,T holds
( f in the carrier of b1 iff f is continuous ) ) & ( for f being Function of S,T holds
( f in the carrier of b2 iff f is continuous ) ) holds
b1 = b2
end;
:: deftheorem Def2 defines SCMaps WAYBEL17:def 2 :
definition
let S be non
empty RelStr ;
let a be
Element of
S,
b be
Element of
S;
func Net-Str c2,
c3 -> non
empty strict NetStr of
a1 means :
Def3:
:: WAYBEL17:def 3
( the
carrier of
it = NAT & the
mapping of
it = a,
b ,... & ( for
i,
j being
Element of
it for
i',
j' being
Element of
NAT st
i = i' &
j = j' holds
(
i <= j iff
i' <= j' ) ) );
existence
ex b1 being non empty strict NetStr of S st
( the carrier of b1 = NAT & the mapping of b1 = a,b ,... & ( for i, j being Element of b1
for i', j' being Element of NAT st i = i' & j = j' holds
( i <= j iff i' <= j' ) ) )
uniqueness
for b1, b2 being non empty strict NetStr of S st the carrier of b1 = NAT & the mapping of b1 = a,b ,... & ( for i, j being Element of b1
for i', j' being Element of NAT st i = i' & j = j' holds
( i <= j iff i' <= j' ) ) & the carrier of b2 = NAT & the mapping of b2 = a,b ,... & ( for i, j being Element of b2
for i', j' being Element of NAT st i = i' & j = j' holds
( i <= j iff i' <= j' ) ) holds
b1 = b2
end;
:: deftheorem Def3 defines Net-Str WAYBEL17:def 3 :
theorem Th5: :: WAYBEL17:5
theorem Th6: :: WAYBEL17:6
theorem Th7: :: WAYBEL17:7
Lemma121:
for S being with_infima Poset
for a, b being Element of S st a <= b holds
lim_inf (Net-Str a,b) = a
theorem Th8: :: WAYBEL17:8
:: deftheorem Def4 defines Net-Str WAYBEL17:def 4 :
theorem Th9: :: WAYBEL17:9
Lemma134:
for R being up-complete LATTICE
for N being reflexive monotone net of R holds lim_inf N = sup N
theorem Th10: :: WAYBEL17:10
theorem Th11: :: WAYBEL17:11
theorem Th12: :: WAYBEL17:12
theorem Th13: :: WAYBEL17:13
theorem Th14: :: WAYBEL17:14
theorem Th15: :: WAYBEL17:15
theorem Th16: :: WAYBEL17:16
theorem Th17: :: WAYBEL17:17
Lemma148:
for S, T being complete LATTICE
for D being Subset of S
for f being Function of S,T st f is monotone holds
f . ("/\" D,S) <= inf (f .: D)
theorem Th18: :: WAYBEL17:18
theorem Th19: :: WAYBEL17:19
Lemma153:
for S, T being complete LATTICE
for f being Function of S,T st ( for N being net of S holds f . (lim_inf N) <= lim_inf (f * N) ) holds
f is directed-sups-preserving
theorem Th20: :: WAYBEL17:20
Lemma161:
for S, T being complete Scott TopLattice
for f being continuous Function of S,T
for N being net of S holds f . (lim_inf N) <= lim_inf (f * N)
Lemma164:
for L being continuous Scott TopLattice
for p being Element of L
for S being Subset of L st S is open & p in S holds
ex q being Element of L st
( q << p & q in S )
Lemma167:
for L being lower-bounded continuous Scott TopLattice
for x being Element of L holds wayabove x is open
Lemma169:
for L being lower-bounded continuous Scott TopLattice
for p being Element of L holds { (wayabove q) where q is Element of L : q << p } is Basis of p
Lemma171:
for T being lower-bounded continuous Scott TopLattice holds { (wayabove x) where x is Element of T : verum } is Basis of T
Lemma172:
for T being lower-bounded continuous Scott TopLattice
for S being Subset of T holds Int S = union { (wayabove x) where x is Element of T : wayabove x c= S }
Lemma173:
for S, T being lower-bounded continuous Scott TopLattice
for f being Function of S,T st ( for x being Element of S
for y being Element of T holds
( y << f . x iff ex w being Element of S st
( w << x & y << f . w ) ) ) holds
f is continuous
theorem Th21: :: WAYBEL17:21
theorem Th22: :: WAYBEL17:22
Lemma179:
for S, T being complete continuous Scott TopLattice
for f being Function of S,T st ( for x being Element of S holds f . x = "\/" { (f . w) where w is Element of S : w << x } ,T ) holds
f is directed-sups-preserving
theorem Th23: :: WAYBEL17:23
theorem Th24: :: WAYBEL17:24
Lemma182:
for S, T being complete Scott TopLattice
for f being Function of S,T st S is algebraic & T is algebraic & ( for x being Element of S
for y being Element of T holds
( y << f . x iff ex w being Element of S st
( w << x & y << f . w ) ) ) holds
for x being Element of S
for k being Element of T st k in the carrier of (CompactSublatt T) holds
( k <= f . x iff ex j being Element of S st
( j in the carrier of (CompactSublatt S) & j <= x & k <= f . j ) )
Lemma184:
for S, T being complete Scott TopLattice
for f being Function of S,T st S is algebraic & T is algebraic & ( for x being Element of S
for k being Element of T st k in the carrier of (CompactSublatt T) holds
( k <= f . x iff ex j being Element of S st
( j in the carrier of (CompactSublatt S) & j <= x & k <= f . j ) ) ) holds
for x being Element of S
for y being Element of T holds
( y << f . x iff ex w being Element of S st
( w << x & y << f . w ) )
Lemma185:
for S, T being complete Scott TopLattice
for f being Function of S,T st S is algebraic & T is algebraic & ( for x being Element of S holds f . x = "\/" { (f . w) where w is Element of S : w << x } ,T ) holds
for x being Element of S holds f . x = "\/" { (f . w) where w is Element of S : ( w <= x & w is compact ) } ,T
theorem Th25: :: WAYBEL17:25
theorem Th26: :: WAYBEL17:26
theorem Th27: :: WAYBEL17:27
theorem Th28: :: WAYBEL17:28
theorem Th29: :: WAYBEL17:29