:: The Equational Characterization of Continuous Lattices
:: by Mariusz \.Zynel
::
:: Received October 25, 1996
:: Copyright (c) 1996-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies WAYBEL_0, ORDINAL2, SUBSET_1, WAYBEL_3, CARD_FIL, XXREAL_0,
TARSKI, XBOOLE_0, YELLOW_0, LATTICE3, LATTICES, YELLOW_2, WAYBEL_1,
YELLOW_1, RELAT_1, FUNCT_1, STRUCT_0, REWRITE1, PBOOLE, FUNCOP_1,
FUNCT_6, CARD_3, FINSEQ_4, ORDERS_2, YELLOW_6, CLASSES1, CLASSES2,
ZFMISC_1, FUNCT_5, EQREL_1, FUNCT_2, FINSUB_1, FINSET_1, ORDERS_1,
RELAT_2, LATTICE2, CAT_1, XBOOLEAN, PARTFUN1, MEMBER_1, SEQM_3, WAYBEL_5,
CARD_1;
notations TARSKI, XBOOLE_0, ZFMISC_1, XTUPLE_0, SUBSET_1, RELAT_1, FINSET_1,
FINSUB_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1, FUNCT_4,
FUNCT_5, FUNCT_6, FUNCOP_1, ORDERS_1, DOMAIN_1, STRUCT_0, ORDERS_2,
LATTICE3, CARD_3, PBOOLE, PRALG_1, PRALG_2, MSUALG_3, CLASSES1, CLASSES2,
YELLOW_0, YELLOW_1, YELLOW_2, WAYBEL_1, WAYBEL_0, WAYBEL_3, YELLOW_6;
constructors FINSUB_1, CLASSES1, CLASSES2, BORSUK_1, LATTICE3, PRALG_1,
PRALG_2, MSUALG_3, ORDERS_3, WAYBEL_1, WAYBEL_3, YELLOW_6, FUNCT_5,
XTUPLE_0;
registrations XBOOLE_0, SUBSET_1, FUNCT_1, RELSET_1, FUNCT_2, FUNCOP_1,
FUNCT_4, FINSET_1, CARD_3, CLASSES2, PBOOLE, STRUCT_0, ORDERS_2,
LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_2, WAYBEL_3, YELLOW_6,
PRALG_1, RELAT_1;
requirements SUBSET, BOOLE;
begin :: The Continuity of Lattices
reserve x, y, i for object,
L for up-complete Semilattice;
theorem :: WAYBEL_5:1 ::Theorem 2.1 (1) iff (2)
L is continuous iff for x being Element of L holds waybelow x is Ideal
of L & x <= sup waybelow x & for I being Ideal of L st x <= sup I holds
waybelow x c= I;
theorem :: WAYBEL_5:2 ::Theorem 2.1 (1) iff (3)
L is continuous iff for x being Element of L ex I being Ideal of L st
x <= sup I & for J being Ideal of L st x <= sup J holds I c= J;
theorem :: WAYBEL_5:3 ::Theorem 2.1 (1) implies (4)
for L being continuous lower-bounded sup-Semilattice holds SupMap L is
upper_adjoint;
theorem :: WAYBEL_5:4 ::Theorem 2.1 (4) implies (1)
for L being up-complete lower-bounded LATTICE holds SupMap L is
upper_adjoint implies L is continuous;
theorem :: WAYBEL_5:5 ::Theorem 2.1 (5) implies (4)
for L being complete Semilattice holds SupMap L is infs-preserving
sups-preserving implies SupMap L is upper_adjoint;
:: Corollary 2.2 can be found in WAYBEL_4.
definition
let J, D be set, K be ManySortedSet of J;
mode DoubleIndexedSet of K, D is ManySortedFunction of K, (J --> D);
end;
definition
let J be set, K be ManySortedSet of J, S be 1-sorted;
mode DoubleIndexedSet of K, S is DoubleIndexedSet of K, the carrier of S;
end;
theorem :: WAYBEL_5:6
for J, D being set, K being ManySortedSet of J for F being
DoubleIndexedSet of K, D for j being set st j in J holds F.j is Function of K.j
, D;
definition
let J, D be non empty set, K be ManySortedSet of J;
let F be DoubleIndexedSet of K, D;
let j be Element of J;
redefine func F.j -> Function of K.j, D;
end;
registration
let J, D be non empty set, K be non-empty ManySortedSet of J;
let F be DoubleIndexedSet of K, D;
let j be Element of J;
cluster rng(F.j) -> non empty;
end;
registration
let J be set, D be non empty set;
let K be non-empty ManySortedSet of J;
cluster -> non-empty for DoubleIndexedSet of K, D;
end;
registration let F be Function-yielding Function;
cluster dom Frege F -> functional;
end;
::$CT
theorem :: WAYBEL_5:8
for F being Function-yielding Function
for f being Function st f in dom Frege F holds
dom f = dom F & dom F = dom((Frege F).f);
theorem :: WAYBEL_5:9
for F being Function-yielding Function
for f being Function st f in dom Frege F
for i being set st i in dom F holds f.i in dom(F.i) &
((Frege F).f).i = (F.i).(f.i) & (F.i).(f.i) in rng((Frege F).f);
theorem :: WAYBEL_5:10
for J, D being set, K being ManySortedSet of J
for F being DoubleIndexedSet of K, D
for f being Function st f in dom(Frege F) holds
(Frege F).f is Function of J, D;
registration
let f be non-empty Function-yielding Function;
cluster doms f -> non-empty;
end;
definition
let J, D be set, K be ManySortedSet of J;
let F be DoubleIndexedSet of K, D;
redefine func Frege F -> DoubleIndexedSet of (product doms F --> J), D;
end;
definition
let J, D be non empty set, K be non-empty ManySortedSet of J;
let F be DoubleIndexedSet of K, D;
let G be DoubleIndexedSet of (product doms F --> J), D;
let f be Element of product doms F;
redefine func G.f -> Function of J, D;
end;
definition
let L be non empty RelStr;
let F be Function-yielding Function;
func \//(F, L) -> Function of dom F, the carrier of L means
:: WAYBEL_5:def 1
for x st x in dom F holds it.x = \\/(F.x, L);
func /\\(F, L) -> Function of dom F, the carrier of L means
:: WAYBEL_5:def 2
for x st x in dom F holds it.x = //\(F.x, L);
end;
notation
let J be set, K be ManySortedSet of J, L be non empty RelStr;
let F be DoubleIndexedSet of K, L;
synonym Sups F for \//(F, L);
synonym Infs F for /\\(F, L);
end;
notation
let I, J be set, L be non empty RelStr;
let F be DoubleIndexedSet of (I --> J), L;
synonym Sups F for \//(F, L);
synonym Infs F for /\\(F, L);
end;
theorem :: WAYBEL_5:11
for L being non empty RelStr for F, G being Function-yielding
Function st dom F = dom G & (for x st x in dom F holds \\/(F.x, L) = \\/(G.x, L
)) holds \//(F, L) = \//(G, L);
theorem :: WAYBEL_5:12
for L being non empty RelStr for F, G being Function-yielding
Function st dom F = dom G & (for x st x in dom F holds //\(F.x, L) = //\(G.x, L
)) holds /\\(F, L) = /\\(G, L);
theorem :: WAYBEL_5:13
for L being non empty RelStr for F being Function-yielding
Function holds (y in rng \//(F, L) iff ex x st x in dom F & y = \\/(F.x, L)) &
(y in rng /\\(F, L) iff ex x st x in dom F & y = //\(F.x, L));
theorem :: WAYBEL_5:14
for L being non empty RelStr for J being non empty set, K being
ManySortedSet of J for F being DoubleIndexedSet of K, L holds (x in rng Sups F
iff ex j being Element of J st x = Sup(F.j)) & (x in rng Infs F iff ex j being
Element of J st x = Inf(F.j));
registration
let J be non empty set, K be ManySortedSet of J, L be non empty RelStr;
let F be DoubleIndexedSet of K, L;
cluster rng Sups F -> non empty;
cluster rng Infs F -> non empty;
end;
reserve L for complete LATTICE,
a, b, c for Element of L,
J for non empty set,
K for non-empty ManySortedSet of J;
theorem :: WAYBEL_5:15
for F being Function-yielding Function holds (for f being
Function st f in dom Frege F holds //\((Frege F).f, L) <= a) implies Sup /\\(
Frege F, L) <= a;
theorem :: WAYBEL_5:16
for F being DoubleIndexedSet of K, L holds Inf Sups F >= Sup Infs Frege F;
theorem :: WAYBEL_5:17
(L is continuous & for c st c << a holds c <= b) implies a <= b;
theorem :: WAYBEL_5:18 ::Theorem 2.3 (2) implies (1)
(for J being non empty set st J in the_universe_of the carrier
of L for K being non-empty ManySortedSet of J st for j being Element of J holds
K.j in the_universe_of the carrier of L for F being DoubleIndexedSet of K, L st
for j being Element of J holds rng(F.j) is directed holds Inf Sups F = Sup Infs
Frege F) implies L is continuous;
theorem :: WAYBEL_5:19 ::Theorem 2.3 (1) iff (2)
L is continuous iff for J, K for F being DoubleIndexedSet of K, L st
for j being Element of J holds rng(F.j) is directed holds Inf Sups F = Sup Infs
Frege F;
definition
let J, K, D be non empty set;
let F be Function of [:J, K:], D;
redefine func curry F -> DoubleIndexedSet of (J --> K), D;
end;
reserve J, K, D for non empty set,
j for Element of J,
k for Element of K;
theorem :: WAYBEL_5:20
for F being Function of [:J, K:], D holds dom curry F = J & dom(
(curry F).j) = K & F.(j, k) = ((curry F).j).k;
theorem :: WAYBEL_5:21 ::Theorem 2.3 (1) iff (2')
L is continuous iff for J, K being non empty set for F being Function
of [:J, K:], the carrier of L st for j being Element of J holds rng((curry F).j
) is directed holds Inf Sups curry F = Sup Infs Frege curry F;
theorem :: WAYBEL_5:22
for F being Function of [:J, K:], the carrier of L for X being
Subset of L st X = {a where a is Element of L: ex f being non-empty
ManySortedSet of J st f in Funcs(J, Fin K) & ex G being DoubleIndexedSet of f,
L st (for j, x st x in f.j holds (G.j).x = F.(j, x)) & a = Inf Sups G} holds
Inf Sups curry F >= sup X;
theorem :: WAYBEL_5:23 ::Theorem 2.3 (1) iff (3)
L is continuous iff for J, K for F being Function of [:J, K:], the
carrier of L for X being Subset of L st X = {a where a is Element of L: ex f
being non-empty ManySortedSet of J st f in Funcs(J, Fin K) & ex G being
DoubleIndexedSet of f, L st (for j, x st x in f.j holds (G.j).x = F.(j, x)) & a
= Inf Sups G} holds Inf Sups curry F = sup X;
begin :: Completely-Distributive Lattices
definition
let L be non empty RelStr;
attr L is completely-distributive means
:: WAYBEL_5:def 3
L is complete & for J being
non empty set, K being non-empty ManySortedSet of J for F being
DoubleIndexedSet of K, L holds Inf Sups F = Sup Infs Frege F;
end;
reserve J for non empty set,
K for non-empty ManySortedSet of J;
registration
cluster -> completely-distributive for 1-element Poset;
end;
registration
cluster completely-distributive for LATTICE;
end;
theorem :: WAYBEL_5:24 ::Corollary 2.5
for L being completely-distributive LATTICE holds L is continuous;
registration
cluster completely-distributive -> complete continuous for LATTICE;
end;
theorem :: WAYBEL_5:25
for L being non empty antisymmetric transitive with_infima
RelStr for x being Element of L for X, Y being Subset of L st ex_sup_of X,L &
ex_sup_of Y,L & Y = {x"/\"y where y is Element of L: y in X} holds x "/\" sup X
>= sup Y;
theorem :: WAYBEL_5:26
for L being completely-distributive LATTICE for X being Subset
of L for x being Element of L holds x "/\" sup X = "\/"({x"/\"y where y is
Element of L: y in X}, L);
registration
cluster completely-distributive -> Heyting for LATTICE;
end;
:: For distributivity confer WAYBEL_1.
begin :: SubFrames of Continuous Lattices
definition
let L be non empty RelStr;
mode CLSubFrame of L is infs-inheriting directed-sups-inheriting non empty
full SubRelStr of L;
end;
theorem :: WAYBEL_5:27
for F being DoubleIndexedSet of K, L st for j being Element of J
holds rng(F.j) is directed holds rng Infs Frege F is directed;
theorem :: WAYBEL_5:28 ::Theorem 2.7 (ii)
L is continuous implies for S being CLSubFrame of L holds S is
continuous LATTICE;
theorem :: WAYBEL_5:29
for S being non empty Poset st ex g being Function of L, S st g
is infs-preserving onto holds S is complete LATTICE;
notation
let J be set, y be set;
synonym J => y for J --> y;
end;
registration
let J be set, y be set;
cluster J => y -> total for J-defined Function;
end;
definition
let A, B, J be set, f be Function of A, B;
redefine func J => f -> ManySortedFunction of (J --> A), (J --> B);
end;
theorem :: WAYBEL_5:30
for A, B being set for f being Function of A, B, g being
Function of B, A st g*f = id A holds (J => g)**(J => f) = id (J --> A);
theorem :: WAYBEL_5:31
for J, A being non empty set, B being set, K being ManySortedSet
of J for F being DoubleIndexedSet of K, A for f being Function of A, B holds (J
=> f)**F is DoubleIndexedSet of K, B;
theorem :: WAYBEL_5:32
for J, A, B being non empty set, K being ManySortedSet of J for
F being DoubleIndexedSet of K, A for f being Function of A, B holds doms((J =>
f)**F) = doms F;
theorem :: WAYBEL_5:33 ::Theorem 2.7 (iii)
L is continuous implies for S being non empty Poset st ex g being
Function of L, S st g is infs-preserving directed-sups-preserving & g is onto
holds S is continuous LATTICE;