Copyright (c) 1996 Association of Mizar Users
environ
vocabulary ORDERS_1, QUANTAL1, RELAT_2, FINSET_1, LATTICE3, BOOLE, SUBSET_1,
LATTICES, YELLOW_0, CAT_1, PRE_TOPC, FUNCT_1, RELAT_1, SEQM_3, FUNCOP_1,
TARSKI, SETFAM_1, ORDINAL2, FILTER_2, FILTER_0, BHSP_3, REALSET1,
WAYBEL_0;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, RELAT_2, RELSET_1,
FINSET_1, FUNCT_1, FUNCT_2, FUNCOP_1, STRUCT_0, REALSET1, PRE_TOPC,
LATTICE3, YELLOW_0, ORDERS_1, ORDERS_3;
constructors LATTICE3, YELLOW_0, ORDERS_3, MEMBERED, PRE_TOPC;
clusters STRUCT_0, FINSET_1, RELSET_1, ORDERS_1, LATTICE3, YELLOW_0, FUNCOP_1,
SUBSET_1, XBOOLE_0;
requirements SUBSET, BOOLE;
definitions TARSKI, STRUCT_0, REALSET1, LATTICE3, YELLOW_0, ORDERS_3,
XBOOLE_0;
theorems TARSKI, ZFMISC_1, ORDERS_1, LATTICE3, FUNCT_2, FUNCOP_1, YELLOW_0,
RELAT_2, FUNCT_1, RELAT_1, PRE_TOPC, ORDERS_3, RELSET_1, SETFAM_1,
XBOOLE_0, XBOOLE_1;
schemes FINSET_1, XBOOLE_0;
begin :: Directed subsets
definition
let L be RelStr;
let X be Subset of L;
attr X is directed means:Def1: :: Definition 1.1
for x,y being Element of L st x in X & y in X
ex z being Element of L st z in X & x <= z & y <= z;
attr X is filtered means:Def2: :: Definition 1.1
for x,y being Element of L st x in X & y in X
ex z being Element of L st z in X & z <= x & z <= y;
end;
:: Original "directed subset" is equivalent to "non empty directed subset"
:: in our terminology. It is shown bellow.
theorem Th1:
for L being non empty transitive RelStr, X being Subset of L holds
X is non empty directed iff
for Y being finite Subset of X
ex x being Element of L st x in X & x is_>=_than Y
proof let L be non empty transitive RelStr, X be Subset of L;
hereby assume X is non empty;
then reconsider X' = X as non empty set;
assume
A1: X is directed;
let Y be finite Subset of X;
defpred P[set] means ex x being Element of L st x in X & x is_>=_than $1;
A2: Y is finite;
consider x being Element of X'; x in X & X c= the carrier of L;
then reconsider x as Element of L;
x is_>=_than {} by YELLOW_0:6; then
A3: P[{}];
A4: now let x,B be set; assume
A5: x in Y & B c= Y;
assume P[B]; then
consider y being Element of L such that
A6: y in X & y is_>=_than B;
x in X by A5;
then reconsider x' = x as Element of L;
consider z being Element of L such that
A7: z in X & x' <= z & y <= z by A1,A5,A6,Def1;
thus P[B \/ {x}]
proof take z; thus z in X by A7;
let a be Element of L;
reconsider a' = a as Element of L;
assume a in B \/ {x};
then a' in B or a' in {x} by XBOOLE_0:def 2;
then y >= a' or a' = x by A6,LATTICE3:def 9,TARSKI:def 1;
hence thesis by A7,ORDERS_1:26;
end;
end;
thus P[Y] from Finite(A2,A3,A4);
end;
assume
A8: for Y being finite Subset of X
ex x being Element of L st x in X & x is_>=_than Y;
{} c= X by XBOOLE_1:2;
then ex x being Element of L st x in X & x is_>=_than {} by A8;
hence X is non empty;
let x,y be Element of L; assume x in X & y in X;
then {x,y} c= X by ZFMISC_1:38;
then consider z being Element of L such that
A9: z in X & z is_>=_than {x,y} by A8;
take z; x in {x,y} & y in {x,y} by TARSKI:def 2;
hence thesis by A9,LATTICE3:def 9;
end;
:: Original "filtered subset" is equivalent to "non empty filtered subset"
:: in our terminology. It is shown bellow.
theorem Th2:
for L being non empty transitive RelStr, X being Subset of L holds
X is non empty filtered iff
for Y being finite Subset of X
ex x being Element of L st x in X & x is_<=_than Y
proof let L be non empty transitive RelStr, X be Subset of L;
hereby assume X is non empty;
then reconsider X' = X as non empty set;
assume
A1: X is filtered;
let Y be finite Subset of X;
defpred P[set] means ex x being Element of L st x in X & x is_<=_than $1;
A2: Y is finite;
consider x being Element of X'; x in X & X c= the carrier of L;
then reconsider x as Element of L;
x is_<=_than {} by YELLOW_0:6; then
A3: P[{}];
A4: now let x,B be set; assume
A5: x in Y & B c= Y;
assume P[B]; then
consider y being Element of L such that
A6: y in X & y is_<=_than B;
x in X by A5;
then reconsider x' = x as Element of L;
consider z being Element of L such that
A7: z in X & x' >= z & y >= z by A1,A5,A6,Def2;
thus P[B \/ {x}]
proof take z; thus z in X by A7;
let a be Element of L;
reconsider a' = a as Element of L;
assume a in B \/ {x};
then a' in B or a' in {x} by XBOOLE_0:def 2;
then y <= a' or a' = x by A6,LATTICE3:def 8,TARSKI:def 1;
hence thesis by A7,ORDERS_1:26;
end;
end;
thus P[Y] from Finite(A2,A3,A4);
end;
assume
A8: for Y being finite Subset of X
ex x being Element of L st x in X & x is_<=_than Y;
{} c= X by XBOOLE_1:2;
then ex x being Element of L st x in X & x is_<=_than {} by A8;
hence X is non empty;
let x,y be Element of L; assume x in X & y in X;
then {x,y} c= X by ZFMISC_1:38;
then consider z being Element of L such that
A9: z in X & z is_<=_than {x,y} by A8;
take z; x in {x,y} & y in {x,y} by TARSKI:def 2;
hence thesis by A9,LATTICE3:def 8;
end;
definition
let L be RelStr;
cluster {}L -> directed filtered;
coherence
proof
thus {}L is directed proof let x be Element of L; thus thesis; end;
let x be Element of L; thus thesis;
end;
end;
definition
let L be RelStr;
cluster directed filtered Subset of L;
existence proof take {}L; thus thesis; end;
end;
theorem Th3:
for L1,L2 being RelStr st the RelStr of L1 = the RelStr of L2
for X1 being Subset of L1, X2 being Subset of L2
st X1 = X2 & X1 is directed
holds X2 is directed
proof let L1,L2 be RelStr such that
A1: the RelStr of L1 = the RelStr of L2;
let X1 be Subset of L1, X2 be Subset of L2 such that
A2: X1 = X2;
assume
A3: for x,y being Element of L1 st x in X1 & y in X1
ex z being Element of L1 st z in X1 & x <= z & y <= z;
let x,y be Element of L2;
reconsider x' = x, y' = y as Element of L1 by A1;
assume x in X2 & y in X2;
then consider z' being Element of L1 such that
A4: z' in X1 & x' <= z' & y' <= z' by A2,A3;
reconsider z = z' as Element of L2 by A1;
take z; thus thesis by A1,A2,A4,YELLOW_0:1;
end;
theorem
for L1,L2 being RelStr st the RelStr of L1 = the RelStr of L2
for X1 being Subset of L1, X2 being Subset of L2
st X1 = X2 & X1 is filtered
holds X2 is filtered
proof let L1,L2 be RelStr such that
A1: the RelStr of L1 = the RelStr of L2;
let X1 be Subset of L1, X2 be Subset of L2 such that
A2: X1 = X2;
assume
A3: for x,y being Element of L1 st x in X1 & y in X1
ex z being Element of L1 st z in X1 & x >= z & y >= z;
let x,y be Element of L2;
reconsider x' = x, y' = y as Element of L1 by A1;
assume x in X2 & y in X2;
then consider z' being Element of L1 such that
A4: z' in X1 & x' >= z' & y' >= z' by A2,A3;
reconsider z = z' as Element of L2 by A1;
take z; thus thesis by A1,A2,A4,YELLOW_0:1;
end;
theorem Th5:
for L being non empty reflexive RelStr, x being Element of L
holds {x} is directed filtered
proof let L be non empty reflexive RelStr; let x be Element of L;
set X = {x};
hereby let z,y be Element of L; assume
A1: z in X & y in X;
take x; thus x in X & z <= x & y <= x by A1,TARSKI:def 1;
end;
hereby let z,y be Element of L; assume
A2: z in X & y in X;
take x; thus x in X & x <= z & x <= y by A2,TARSKI:def 1;
end;
end;
definition
let L be non empty reflexive RelStr;
cluster directed filtered non empty finite Subset of L;
existence
proof consider x being Element of L;
take {x}; thus thesis by Th5;
end;
end;
definition
let L be with_suprema RelStr;
cluster [#]L -> directed;
coherence
proof set X = [#]L;
A1: X = the carrier of L by PRE_TOPC:12;
let x,y be Element of L; assume x in X & y in X;
ex z being Element of L st x <= z & y <= z &
for z' being Element of L st x <= z' & y <= z' holds z <= z'
by LATTICE3:def 10;
hence thesis by A1;
end;
end;
definition
let L be upper-bounded non empty RelStr;
cluster [#]L -> directed;
coherence
proof set X = [#]L;
A1: X = the carrier of L by PRE_TOPC:12;
let x,y be Element of L; assume x in X & y in X;
consider z being Element of L such that
A2: z is_>=_than X by A1,YELLOW_0:def 5;
take z; thus thesis by A1,A2,LATTICE3:def 9;
end;
end;
definition
let L be with_infima RelStr;
cluster [#]L -> filtered;
coherence
proof set X = [#]L;
A1: X = the carrier of L by PRE_TOPC:12;
let x,y be Element of L such that x in X & y in X;
ex z being Element of L st z <= x & z <= y &
for z' being Element of L st z' <= x & z' <= y holds z' <= z
by LATTICE3:def 11;
hence thesis by A1;
end;
end;
definition
let L be lower-bounded non empty RelStr;
cluster [#]L -> filtered;
coherence
proof set X = [#]L;
A1: X = the carrier of L by PRE_TOPC:12;
let x,y be Element of L; assume x in X & y in X;
consider z being Element of L such that
A2: z is_<=_than X by A1,YELLOW_0:def 4;
take z; thus thesis by A1,A2,LATTICE3:def 8;
end;
end;
definition
let L be non empty RelStr;
let S be SubRelStr of L;
attr S is filtered-infs-inheriting means:Def3:
for X being filtered Subset of S st X <> {} & ex_inf_of X,L
holds "/\"(X,L) in the carrier of S;
attr S is directed-sups-inheriting means:Def4:
for X being directed Subset of S st X <> {} & ex_sup_of X,L
holds "\/"(X,L) in the carrier of S;
end;
definition
let L be non empty RelStr;
cluster infs-inheriting -> filtered-infs-inheriting SubRelStr of L;
coherence
proof let S be SubRelStr of L such that
A1: for X being Subset of S st ex_inf_of X,L holds "/\"(X,L) in
the carrier of S;
let X be filtered Subset of S;
thus thesis by A1;
end;
cluster sups-inheriting -> directed-sups-inheriting SubRelStr of L;
coherence
proof let S be SubRelStr of L such that
A2: for X being Subset of S st ex_sup_of X,L holds "\/"(X,L) in
the carrier of S;
let X be directed Subset of S;
thus thesis by A2;
end;
end;
definition
let L be non empty RelStr;
cluster infs-inheriting sups-inheriting non empty full strict SubRelStr of L;
existence
proof
consider S being infs-inheriting sups-inheriting non empty full
strict SubRelStr of L;
take S; thus thesis;
end;
end;
theorem
for L being non empty transitive RelStr
for S being filtered-infs-inheriting non empty full SubRelStr of L
for X being filtered Subset of S st X <> {} & ex_inf_of X,L
holds ex_inf_of X,S & "/\"(X,S) = "/\"(X,L)
proof let L be non empty transitive RelStr;
let S be filtered-infs-inheriting non empty full SubRelStr of L;
let X be filtered Subset of S; assume
A1: X <> {} & ex_inf_of X,L;
then "/\"(X,L) in the carrier of S by Def3;
hence thesis by A1,YELLOW_0:64;
end;
theorem
for L being non empty transitive RelStr
for S being directed-sups-inheriting non empty full SubRelStr of L
for X being directed Subset of S st X <> {} & ex_sup_of X,L
holds ex_sup_of X,S & "\/"(X,S) = "\/"(X,L)
proof let L be non empty transitive RelStr;
let S be directed-sups-inheriting non empty full SubRelStr of L;
let X be directed Subset of S; assume
A1: X <> {} & ex_sup_of X,L;
then "\/"(X,L) in the carrier of S by Def4;
hence thesis by A1,YELLOW_0:65;
end;
begin :: Nets
definition
let L1,L2 be non empty 1-sorted;
let f be map of L1,L2;
let x be Element of L1;
redefine func f.x -> Element of L2;
coherence
proof f.x in the carrier of L2;
hence thesis;
end;
end;
definition
let L1,L2 be RelStr;
let f be map of L1,L2;
attr f is antitone means
for x,y being Element of L1 st x <= y
for a,b being Element of L2 st a = f.x & b = f.y holds a >= b;
end;
:: Definition 1.2
definition let L be 1-sorted;
struct (RelStr) NetStr over L (#
carrier -> set,
InternalRel -> (Relation of the carrier),
mapping -> Function of the carrier, the carrier of L
#);
end;
definition let L be 1-sorted;
let X be non empty set;
let O be Relation of X;
let F be Function of X, the carrier of L;
cluster NetStr(#X,O,F#) -> non empty;
coherence proof thus the carrier of NetStr(#X,O,F#) is non empty; end;
end;
definition
let N be RelStr;
attr N is directed means:Def6:
[#]N is directed;
end;
definition let L be 1-sorted;
cluster non empty reflexive transitive antisymmetric directed
(strict NetStr over L);
existence
proof consider X being with_suprema Poset;
consider M being Function of the carrier of X, the carrier of L;
take N = NetStr(#the carrier of X, the InternalRel of X, M#);
thus N is non empty;
the InternalRel of N is_reflexive_in the carrier of N &
the InternalRel of N is_transitive_in the carrier of N &
the InternalRel of N is_antisymmetric_in the carrier of N
by ORDERS_1:def 4,def 5,def 6;
hence N is reflexive transitive antisymmetric
by ORDERS_1:def 4,def 5,def 6;
A1: the RelStr of N = the RelStr of X;
[#]X = the carrier of X by PRE_TOPC:12 .= [#]N by PRE_TOPC:12;
hence [#]N is directed by A1,Th3;
end;
end;
definition let L be 1-sorted;
mode prenet of L is directed non empty NetStr over L;
end;
definition let L be 1-sorted;
mode net of L is transitive prenet of L;
end;
definition
let L be non empty 1-sorted;
let N be non empty NetStr over L;
func netmap(N,L) -> map of N,L equals:Def7:
the mapping of N;
coherence;
let i be Element of N;
func N.i -> Element of L equals:Def8:
(the mapping of N).i;
correctness;
end;
definition
let L be non empty RelStr;
let N be non empty NetStr over L;
attr N is monotone means
netmap(N,L) is monotone;
attr N is antitone means
netmap(N,L) is antitone;
end;
definition let L be non empty 1-sorted;
let N be non empty NetStr over L;
let X be set;
pred N is_eventually_in X means:Def11:
ex i being Element of N st
for j being Element of N st i <= j holds N.j in X;
pred N is_often_in X means
for i being Element of N
ex j being Element of N st i <= j & N.j in X;
end;
theorem
for L being non empty 1-sorted, N being non empty NetStr over L
for X,Y being set st X c= Y holds
(N is_eventually_in X implies N is_eventually_in Y) &
(N is_often_in X implies N is_often_in Y)
proof let L be non empty 1-sorted, N be non empty NetStr over L;
let X,Y be set such that
A1: X c= Y;
hereby assume N is_eventually_in X;
then consider i being Element of N such that
A2: for j being Element of N st i <= j holds N.j in X by Def11;
thus N is_eventually_in Y
proof take i; let j be Element of N; assume i <= j;
then N.j in X by A2;
hence thesis by A1;
end;
end;
assume
A3: for i being Element of N ex j being Element of N st i <= j & N.j in X;
let i be Element of N; consider j being Element of N such that
A4: i <= j & N.j in X by A3;
take j; thus thesis by A1,A4;
end;
theorem
for L being non empty 1-sorted, N being non empty NetStr over L
for X being set holds
N is_eventually_in X iff not N is_often_in (the carrier of L) \ X
proof let L be non empty 1-sorted, N be non empty NetStr over L, X be set;
set Y = (the carrier of L) \ X;
thus N is_eventually_in X implies not N is_often_in Y
proof given i being Element of N such that
A1: for j being Element of N st i <= j holds N.j in X;
take i; let j be Element of N;
assume i <= j;
then N.j in X by A1;
hence thesis by XBOOLE_0:def 4;
end;
given i being Element of N such that
A2: for j being Element of N st i <= j holds not N.j in Y;
take i; let j be Element of N;
assume i <= j;
then not N.j in Y by A2;
hence thesis by XBOOLE_0:def 4;
end;
theorem
for L being non empty 1-sorted, N being non empty NetStr over L
for X being set holds
N is_often_in X iff not N is_eventually_in (the carrier of L) \ X
proof let L be non empty 1-sorted, N be non empty NetStr over L, X be set;
set Y = (the carrier of L) \ X;
thus N is_often_in X implies not N is_eventually_in Y
proof assume
A1: for i being Element of N ex j being Element of N st i <= j & N.j in X;
let i be Element of N; consider j being Element of N such that
A2: i <= j & N.j in X by A1;
take j; thus thesis by A2,XBOOLE_0:def 4;
end;
assume
A3: for i being Element of N ex j being Element of N st i <= j & not N.j in Y;
let i be Element of N; consider j being Element of N such that
A4: i <= j & not N.j in Y by A3;
take j; thus thesis by A4,XBOOLE_0:def 4;
end;
Holds_Eventually
{L() -> non empty RelStr, N() -> (non empty NetStr over L()), P[set]}:
N() is_eventually_in
{N().k where k is Element of N(): P[N().k]} iff
ex i being Element of N() st
for j being Element of N() st i <= j holds P[N().j]
provided for x being Element of L() holds P[x] iff P[x]
proof
set X = {N().k where k is Element of N(): P[N().k]};
hereby assume N() is_eventually_in X;
then consider i being Element of N() such that
A1: for j being Element of N() st i <= j holds N().j in X by Def11;
take i; let j be Element of N(); assume i <= j;
then N().j in X by A1;
then ex k being Element of N() st N().j = N().k & P[N().k];
hence P[N().j];
end;
given i being Element of N() such that
A2: for j being Element of N() st i <= j holds P[N().j];
take i; let j be Element of N(); assume i <= j;
then P[N().j] by A2;
hence N().j in X;
end;
definition
let L be non empty RelStr;
let N be non empty NetStr over L;
attr N is eventually-directed means:Def13:
for i being Element of N holds N is_eventually_in
{N.j where j is Element of N: N.i <= N.j};
attr N is eventually-filtered means:Def14:
for i being Element of N holds N is_eventually_in
{N.j where j is Element of N: N.i >= N.j};
end;
theorem Th11:
for L being non empty RelStr, N being non empty NetStr over L holds
N is eventually-directed
iff
for i being Element of N ex j being Element of N st
for k being Element of N st j <= k holds N.i <= N.k
proof let L be non empty RelStr, N be non empty NetStr over L;
A1: now let i be Element of N;
defpred P[Element of L] means N.i <= $1;
A2: for x being Element of L holds P[x] iff P[x];
thus N is_eventually_in
{N.j where j is Element of N: P[N.j]} iff
ex k being Element of N st
for l being Element of N st k <= l holds P[N.l]
from Holds_Eventually(A2);
end;
hereby assume
A3: N is eventually-directed;
let i be Element of N;
N is_eventually_in
{N.j where j is Element of N: N.i <= N.j} by A3,Def13;
hence ex j being Element of N st
for k being Element of N st j <= k holds N.i <= N.k
by A1;
end;
assume
A4: for i being Element of N ex j being Element of N st
for k being Element of N st j <= k holds N.i <= N.k;
let i be Element of N;
ex j being Element of N st
for k being Element of N st j <= k holds N.i <= N.k
by A4;
hence thesis by A1;
end;
theorem Th12:
for L being non empty RelStr, N being non empty NetStr over L holds
N is eventually-filtered
iff
for i being Element of N ex j being Element of N st
for k being Element of N st j <= k holds N.i >= N.k
proof let L be non empty RelStr, N be non empty NetStr over L;
A1: now let i be Element of N;
defpred P[Element of L] means N.i >= $1;
A2: for x being Element of L holds P[x] iff P[x];
thus N is_eventually_in
{N.j where j is Element of N: P[N.j]} iff
ex k being Element of N st for l being Element of N st k <=
l holds P[N.l]
from Holds_Eventually(A2);
end;
hereby assume
A3: N is eventually-filtered;
let i be Element of N;
N is_eventually_in {N.j where j is Element of N: N.i >= N.j} by A3,Def14
;
hence ex j being Element of N st
for k being Element of N st j <= k holds N.i >= N.k by A1;
end;
assume
A4: for i being Element of N ex j being Element of N st
for k being Element of N st j <= k holds N.i >= N.k;
let i be Element of N;
ex j being Element of N st
for k being Element of N st j <= k holds N.i >= N.k by A4;
hence thesis by A1;
end;
definition
let L be non empty RelStr;
cluster monotone -> eventually-directed prenet of L;
coherence
proof let N be prenet of L; assume
A1: for i,j being Element of N st i <= j
for a,b being Element of L st a = netmap(N,L).i & b = netmap(N,L).j
holds a <= b;
A2: netmap(N,L) = the mapping of N by Def7;
now let i be Element of N; take j = i;
let k be Element of N; assume j <= k;
then netmap(N,L).i <= netmap(N,L).k by A1;
then N.i <= netmap(N,L).k by A2,Def8;
hence N.i <= N.k by A2,Def8;
end;
hence thesis by Th11;
end;
cluster antitone -> eventually-filtered prenet of L;
coherence
proof let N be prenet of L; assume
A3: for i,j being Element of N st i <= j
for a,b being Element of L st a = netmap(N,L).i & b = netmap(N,L).j
holds a >= b;
A4: netmap(N,L) = the mapping of N by Def7;
now let i be Element of N; take j = i;
let k be Element of N; assume j <= k;
then netmap(N,L).i >= netmap(N,L).k by A3;
then N.i >= netmap(N,L).k by A4,Def8;
hence N.i >= N.k by A4,Def8;
end;
hence thesis by Th12;
end;
end;
definition
let L be non empty reflexive RelStr;
cluster monotone antitone strict prenet of L;
existence
proof consider J being upper-bounded (non empty Poset);
consider x being Element of L;
set M = (the carrier of J) --> x;
rng M c= {x} & {x} c= the carrier of L by FUNCOP_1:19;
then dom M = the carrier of J & rng M c= the carrier of L
by FUNCOP_1:19,XBOOLE_1:1;
then reconsider M as Function of the carrier of J, the carrier of L
by FUNCT_2:def 1,RELSET_1:11;
set N = NetStr(#the carrier of J, the InternalRel of J, M#);
A1: the RelStr of N = the RelStr of J;
[#]J = the carrier of J by PRE_TOPC:12 .= [#]N by PRE_TOPC:12;
then [#]N is directed by A1,Th3;
then reconsider N as prenet of L by Def6;
take N;
A2: netmap(N,L) = M by Def7;
thus N is monotone
proof let i,j be Element of N such that i <= j;
let a,b be Element of L; assume
a = netmap(N,L).i & b = netmap(N,L).j;
then a = x & b = x & x <= x by A2,FUNCOP_1:13;
hence thesis;
end;
thus N is antitone
proof let i,j be Element of N such that i <= j;
let a,b be Element of L; assume
a = netmap(N,L).i & b = netmap(N,L).j;
then a = x & b = x & x <= x by A2,FUNCOP_1:13;
hence thesis;
end;
thus thesis;
end;
end;
begin :: Closure by lower elements and finite sups
:: Definition 1.3
definition
let L be RelStr;
let X be Subset of L;
func downarrow X -> Subset of L means:Def15:
for x being Element of L holds x in it iff
ex y being Element of L st y >= x & y in X;
existence
proof
defpred P[set] means ex a,y being Element of L st a = $1 & y >= a & y in X;
consider S being set such that
A1: for x being set holds x in S iff x in the carrier of L & P[x]
from Separation;
S c= the carrier of L proof let x be set; thus thesis by A1; end;
then reconsider S as Subset of L;
take S; let x be Element of L;
hereby assume x in S;
then ex a,y being Element of L st a = x & y >= a & y in X by A1;
hence ex y being Element of L st y >= x & y in X;
end;
thus thesis by A1;
end;
uniqueness
proof let S1,S2 be Subset of L such that
A2: for x being Element of L holds x in S1 iff
ex y being Element of L st y >= x & y in X and
A3: for x being Element of L holds x in S2 iff
ex y being Element of L st y >= x & y in X;
thus S1 c= S2
proof let x be set; assume
A4: x in S1;
then reconsider x as Element of L;
ex y being Element of L st y >= x & y in X by A2,A4;
hence thesis by A3;
end;
let x be set; assume
A5: x in S2;
then reconsider x as Element of L;
ex y being Element of L st y >= x & y in X by A3,A5;
hence thesis by A2;
end;
func uparrow X -> Subset of L means:Def16:
for x being Element of L holds x in it iff
ex y being Element of L st y <= x & y in X;
existence
proof
defpred P[set] means ex a,y being Element of L st a = $1 & y <= a & y in X;
consider S being set such that
A6: for x being set holds x in S iff x in the carrier of L & P[x]
from Separation;
S c= the carrier of L proof let x be set; thus thesis by A6; end;
then reconsider S as Subset of L;
take S; let x be Element of L;
hereby assume x in S;
then ex a,y being Element of L st a = x & y <= a & y in X by A6;
hence ex y being Element of L st y <= x & y in X;
end;
thus thesis by A6;
end;
correctness
proof let S1,S2 be Subset of L such that
A7: for x being Element of L holds x in S1 iff
ex y being Element of L st y <= x & y in X and
A8: for x being Element of L holds x in S2 iff
ex y being Element of L st y <= x & y in X;
thus S1 c= S2
proof let x be set; assume
A9: x in S1;
then reconsider x as Element of L;
ex y being Element of L st y <= x & y in X by A7,A9;
hence thesis by A8;
end;
let x be set; assume
A10: x in S2;
then reconsider x as Element of L;
ex y being Element of L st y <= x & y in X by A8,A10;
hence thesis by A7;
end;
end;
theorem Th13:
for L1,L2 being RelStr st the RelStr of L1 = the RelStr of L2
for X being Subset of L1
for Y being Subset of L2
st X = Y holds downarrow X = downarrow Y & uparrow X = uparrow Y
proof let L1,L2 be RelStr such that
A1: the RelStr of L1 = the RelStr of L2;
let X be Subset of L1;
let Y be Subset of L2 such that
A2: X = Y;
thus downarrow X c= downarrow Y
proof let x be set; assume
A3: x in downarrow X;
then reconsider x as Element of L1;
reconsider x' = x as Element of L2 by A1;
consider y being Element of L1 such that
A4: y >= x & y in X by A3,Def15;
reconsider y' = y as Element of L2 by A1;
y' >= x' by A1,A4,YELLOW_0:1;
hence thesis by A2,A4,Def15;
end;
thus downarrow Y c= downarrow X
proof let x be set; assume
A5: x in downarrow Y;
then reconsider x as Element of L2;
reconsider x' = x as Element of L1 by A1;
consider y being Element of L2 such that
A6: y >= x & y in Y by A5,Def15;
reconsider y' = y as Element of L1 by A1;
y' >= x' by A1,A6,YELLOW_0:1;
hence thesis by A2,A6,Def15;
end;
thus uparrow X c= uparrow Y
proof let x be set; assume
A7: x in uparrow X;
then reconsider x as Element of L1;
reconsider x' = x as Element of L2 by A1;
consider y being Element of L1 such that
A8: y <= x & y in X by A7,Def16;
reconsider y' = y as Element of L2 by A1;
y' <= x' by A1,A8,YELLOW_0:1;
hence thesis by A2,A8,Def16;
end;
thus uparrow Y c= uparrow X
proof let x be set; assume
A9: x in uparrow Y;
then reconsider x as Element of L2;
reconsider x' = x as Element of L1 by A1;
consider y being Element of L2 such that
A10: y <= x & y in Y by A9,Def16;
reconsider y' = y as Element of L1 by A1;
y' <= x' by A1,A10,YELLOW_0:1;
hence thesis by A2,A10,Def16;
end;
end;
theorem Th14:
for L being non empty RelStr, X being Subset of L holds
downarrow X =
{x where x is Element of L: ex y being Element of L st x <= y & y in X}
proof let L be non empty RelStr, X be Subset of L;
set Y={x where x is Element of L: ex y being Element of L st x <=
y & y in X};
hereby let x be set; assume
A1: x in downarrow X;
then reconsider y = x as Element of L;
ex z being Element of L st z >= y & z in X by A1,Def15;
hence x in Y;
end;
let x be set; assume x in Y;
then ex a being Element of L st a = x &
ex y being Element of L st a <= y & y in X;
hence x in downarrow X by Def15;
end;
theorem Th15:
for L being non empty RelStr, X being Subset of L holds
uparrow X =
{x where x is Element of L: ex y being Element of L st x >= y & y in X}
proof let L be non empty RelStr, X be Subset of L;
set Y={x where x is Element of L: ex y being Element of L st x >=
y & y in X};
hereby let x be set; assume
A1: x in uparrow X;
then reconsider y = x as Element of L;
ex z being Element of L st z <= y & z in X by A1,Def16;
hence x in Y;
end;
let x be set; assume x in Y;
then ex a being Element of L st a = x &
ex y being Element of L st a >= y & y in X;
hence x in uparrow X by Def16;
end;
definition
let L be non empty reflexive RelStr;
let X be non empty Subset of L;
cluster downarrow X -> non empty;
coherence
proof consider x being Element of X;
reconsider x' = x as Element of L;
x' <= x';
hence thesis by Def15;
end;
cluster uparrow X -> non empty;
coherence
proof consider x being Element of X;
reconsider x' = x as Element of L;
x' <= x';
hence thesis by Def16;
end;
end;
theorem Th16:
for L being reflexive RelStr, X being Subset of L holds
X c= downarrow X & X c= uparrow X
proof let L be reflexive RelStr, X be Subset of L;
A1: the InternalRel of L is_reflexive_in the carrier of L by ORDERS_1:def 4;
hereby let x be set; assume
A2: x in X;
then reconsider y = x as Element of L;
[y,y] in the InternalRel of L by A1,A2,RELAT_2:def 1;
then y <= y by ORDERS_1:def 9;
hence x in downarrow X by A2,Def15;
end;
let x be set; assume
A3: x in X;
then reconsider y = x as Element of L;
[y,y] in the InternalRel of L by A1,A3,RELAT_2:def 1;
then y <= y by ORDERS_1:def 9;
hence x in uparrow X by A3,Def16;
end;
definition
let L be non empty RelStr;
let x be Element of L;
func downarrow x -> Subset of L equals:Def17: :: Definition 1.3 (iii)
downarrow {x};
correctness;
func uparrow x -> Subset of L equals:Def18: :: Definition 1.3 (iv)
uparrow {x};
correctness;
end;
theorem Th17:
for L being non empty RelStr, x,y being Element of L
holds y in downarrow x iff y <= x
proof let L be non empty RelStr, x,y be Element of L;
downarrow x = downarrow {x} by Def17;
then A1: downarrow x = {z where z is Element of L:
ex v being Element of L st z <= v & v in {x}} by Th14;
then y in downarrow x iff ex z being Element of L st y = z &
ex v being Element of L st z <= v & v in {x};
hence y in downarrow x implies y <= x by TARSKI:def 1;
x in {x} by TARSKI:def 1;
hence thesis by A1;
end;
theorem Th18:
for L being non empty RelStr, x,y being Element of L
holds y in uparrow x iff x <= y
proof let L be non empty RelStr, x,y be Element of L;
uparrow x = uparrow {x} by Def18;
then A1: uparrow x = {z where z is Element of L:
ex v being Element of L st z >= v & v in {x}} by Th15;
then y in uparrow x iff ex z being Element of L st y = z &
ex v being Element of L st z >= v & v in {x};
hence y in uparrow x implies y >= x by TARSKI:def 1;
x in {x} by TARSKI:def 1;
hence thesis by A1;
end;
theorem
for L being non empty reflexive antisymmetric RelStr
for x,y being Element of L
st downarrow x = downarrow y holds x = y
proof let L be non empty reflexive antisymmetric RelStr;
let x,y be Element of L;
reconsider x' = x, y' = y as Element of L;
A1: x' <= x' & y' <= y';
assume
downarrow x = downarrow y;
then y in downarrow x & x in downarrow y by A1,Th17;
then x' <= y' & x' >= y' by Th17;
hence thesis by ORDERS_1:25;
end;
theorem
for L being non empty reflexive antisymmetric RelStr
for x,y being Element of L
st uparrow x = uparrow y holds x = y
proof let L be non empty reflexive antisymmetric RelStr;
let x,y be Element of L;
reconsider x' = x, y' = y as Element of L;
A1: x' <= x' & y' <= y';
assume
uparrow x = uparrow y;
then y in uparrow x & x in uparrow y by A1,Th18;
then x' <= y' & x' >= y' by Th18;
hence thesis by ORDERS_1:25;
end;
theorem Th21:
for L being non empty transitive RelStr
for x,y being Element of L st x <= y
holds downarrow x c= downarrow y
proof let L be non empty transitive RelStr;
let x,y be Element of L such that
A1: x <= y;
let z be set; assume
A2: z in downarrow x;
then reconsider z as Element of L;
z <= x by A2,Th17;
then z <= y by A1,ORDERS_1:26;
hence thesis by Th17;
end;
theorem Th22:
for L being non empty transitive RelStr
for x,y being Element of L st x <= y
holds uparrow y c= uparrow x
proof let L be non empty transitive RelStr;
let x,y be Element of L such that
A1: x <= y;
let z be set; assume
A2: z in uparrow y;
then reconsider z as Element of L;
y <= z by A2,Th18;
then x <= z by A1,ORDERS_1:26;
hence thesis by Th18;
end;
definition
let L be non empty reflexive RelStr;
let x be Element of L;
cluster downarrow x -> non empty directed;
coherence
proof reconsider x' = x as Element of L;
x' <= x';
hence downarrow x is non empty by Th17;
let z,y be Element of L; assume
A1: z in downarrow x & y in downarrow x;
take x'; x' <= x';
hence x' in downarrow x & z <= x' & y <= x' by A1,Th17;
end;
cluster uparrow x -> non empty filtered;
coherence
proof reconsider x' = x as Element of L;
x' <= x';
hence uparrow x is non empty by Th18;
let z,y be Element of L; assume
A2: z in uparrow x & y in uparrow x;
take x'; x' <= x';
hence x' in uparrow x & x' <= z & x' <= y by A2,Th18;
end;
end;
definition
let L be RelStr;
let X be Subset of L;
attr X is lower means:Def19: :: Definition 1.3 (v)
for x,y being Element of L st x in X & y <= x holds y in X;
attr X is upper means:Def20: :: Definition 1.3 (vi)
for x,y being Element of L st x in X & x <= y holds y in X;
end;
definition
let L be RelStr;
cluster lower upper Subset of L;
existence
proof the carrier of L c= the carrier of L;
then reconsider S = the carrier of L as Subset of L;
take S;
thus S is lower proof let x be Element of L; thus thesis; end;
let x be Element of L; thus thesis;
end;
end;
theorem Th23:
for L being RelStr, X being Subset of L holds
X is lower iff downarrow X c= X
proof let L be RelStr, X be Subset of L;
hereby assume
A1: X is lower;
thus downarrow X c= X
proof let x be set; assume
A2: x in downarrow X;
then reconsider x' = x as Element of L;
ex y being Element of L st x' <= y & y in X by A2,Def15;
hence x in X by A1,Def19;
end;
end;
assume
A3: downarrow X c= X;
let x,y be Element of L; assume x in X & y <= x;
then y in downarrow X by Def15;
hence y in X by A3;
end;
theorem Th24:
for L being RelStr, X being Subset of L holds
X is upper iff uparrow X c= X
proof let L be RelStr, X be Subset of L;
hereby assume
A1: X is upper;
thus uparrow X c= X
proof let x be set; assume
A2: x in uparrow X;
then reconsider x' = x as Element of L;
ex y being Element of L st x' >= y & y in X by A2,Def16;
hence x in X by A1,Def20;
end;
end;
assume
A3: uparrow X c= X;
let x,y be Element of L; assume x in X & y >= x;
then y in uparrow X by Def16;
hence y in X by A3;
end;
theorem
for L1,L2 being RelStr st the RelStr of L1 = the RelStr of L2
for X1 being Subset of L1, X2 being Subset of L2
st X1 = X2 holds
(X1 is lower implies X2 is lower) &
(X1 is upper implies X2 is upper)
proof let L1,L2 be RelStr such that
A1: the RelStr of L1 = the RelStr of L2;
let X1 be Subset of L1, X2 be Subset of L2; assume
A2: X1 = X2;
hereby assume X1 is lower;
then downarrow X1 = downarrow X2 & downarrow X1 c= X1 by A1,A2,Th13,Th23;
hence X2 is lower by A2,Th23;
end;
assume X1 is upper;
then uparrow X1 = uparrow X2 & uparrow X1 c= X1 by A1,A2,Th13,Th24;
hence X2 is upper by A2,Th24;
end;
theorem
for L being RelStr, A being Subset of bool the carrier of L st
for X being Subset of L st X in A holds X is lower
holds union A is lower Subset of L
proof let L be RelStr, A be Subset of bool the carrier of L such that
A1: for X being Subset of L st X in A holds X is lower;
reconsider A as Subset-Family of L by SETFAM_1:def 7;
reconsider X = union A as Subset of L;
X is lower
proof let x,y be Element of L; assume x in X;
then consider Y being set such that
A2: x in Y & Y in A by TARSKI:def 4;
reconsider Y as Subset of L by A2;
A3: Y is lower by A1,A2;
assume y <= x;
then y in Y by A2,A3,Def19;
hence thesis by A2,TARSKI:def 4;
end;
hence thesis;
end;
theorem Th27:
for L being RelStr, X,Y being Subset of L st X is lower & Y is lower
holds X /\ Y is lower & X \/ Y is lower
proof let L be RelStr; let X,Y be Subset of L such that
A1: for x,y being Element of L st x in X & y <= x holds y in X and
A2: for x,y being Element of L st x in Y & y <= x holds y in Y;
hereby let x,y be Element of L; assume x in X /\ Y;
then A3: x in X & x in Y by XBOOLE_0:def 3; assume
y <= x;
then y in X & y in Y by A1,A2,A3;
hence y in X /\ Y by XBOOLE_0:def 3;
end;
let x,y be Element of L; assume x in X \/ Y;
then A4: x in X or x in Y by XBOOLE_0:def 2; assume
y <= x;
then y in X or y in Y by A1,A2,A4;
hence y in X \/ Y by XBOOLE_0:def 2;
end;
theorem
for L being RelStr, A being Subset of bool the carrier of L st
for X being Subset of L st X in A holds X is upper
holds union A is upper Subset of L
proof let L be RelStr, A be Subset of bool the carrier of L such that
A1: for X being Subset of L st X in A holds X is upper;
reconsider A as Subset-Family of L by SETFAM_1:def 7;
reconsider X = union A as Subset of L;
X is upper
proof let x,y be Element of L; assume x in X;
then consider Y being set such that
A2: x in Y & Y in A by TARSKI:def 4;
reconsider Y as Subset of L by A2;
A3: Y is upper by A1,A2;
assume y >= x;
then y in Y by A2,A3,Def20;
hence thesis by A2,TARSKI:def 4;
end;
hence thesis;
end;
theorem Th29:
for L being RelStr, X,Y being Subset of L st X is upper & Y is upper
holds X /\ Y is upper & X \/ Y is upper
proof let L be RelStr; let X,Y be Subset of L such that
A1: for x,y being Element of L st x in X & y >= x holds y in X and
A2: for x,y being Element of L st x in Y & y >= x holds y in Y;
hereby let x,y be Element of L; assume x in X /\ Y;
then A3: x in X & x in Y by XBOOLE_0:def 3; assume
y >= x;
then y in X & y in Y by A1,A2,A3;
hence y in X /\ Y by XBOOLE_0:def 3;
end;
let x,y be Element of L; assume x in X \/ Y;
then A4: x in X or x in Y by XBOOLE_0:def 2; assume
y >= x;
then y in X or y in Y by A1,A2,A4;
hence y in X \/ Y by XBOOLE_0:def 2;
end;
definition
let L be non empty transitive RelStr;
let X be Subset of L;
cluster downarrow X -> lower;
coherence
proof let y,z be Element of L; assume y in downarrow X;
then consider x being Element of L such that
A1: y <= x & x in X by Def15;
assume z <= y; then z <= x by A1,ORDERS_1:26;
hence z in downarrow X by A1,Def15;
end;
cluster uparrow X -> upper;
coherence
proof let y,z be Element of L; assume y in uparrow X;
then consider x being Element of L such that
A2: y >= x & x in X by Def16;
assume z >= y; then z >= x by A2,ORDERS_1:26;
hence z in uparrow X by A2,Def16;
end;
end;
definition
let L be non empty transitive RelStr;
let x be Element of L;
cluster downarrow x -> lower;
coherence
proof downarrow x = downarrow {x} by Def17;
hence thesis;
end;
cluster uparrow x -> upper;
coherence
proof uparrow x = uparrow {x} by Def18;
hence thesis;
end;
end;
definition
let L be non empty RelStr;
cluster [#]L -> lower upper;
coherence
proof set X = [#]L;
A1: X = the carrier of L by PRE_TOPC:12;
hence for x,y being Element of L st x in X & y <= x holds y in X;
thus for x,y being Element of L st x in X & x <= y holds y in X by A1;
end;
end;
definition
let L be non empty RelStr;
cluster non empty lower upper Subset of L;
existence proof take [#]L; thus thesis; end;
end;
definition
let L be non empty reflexive transitive RelStr;
cluster non empty lower directed Subset of L;
existence
proof consider x being Element of L;
take downarrow x; thus thesis;
end;
cluster non empty upper filtered Subset of L;
existence
proof consider x being Element of L;
take uparrow x; thus thesis;
end;
end;
definition
let L be with_infima with_suprema Poset;
cluster non empty directed filtered lower upper Subset of L;
existence proof take [#]L; thus thesis; end;
end;
theorem Th30:
for L being non empty transitive reflexive RelStr, X be Subset of L holds
X is directed iff downarrow X is directed
proof let L be non empty transitive reflexive RelStr, X be Subset of L;
thus X is directed implies downarrow X is directed
proof assume
A1: for x,y being Element of L st x in X & y in X
ex z being Element of L st z in X & x <= z & y <= z;
let x,y be Element of L; assume
A2: x in downarrow X & y in downarrow X;
then consider x' being Element of L such that
A3: x <= x' & x' in X by Def15;
consider y' being Element of L such that
A4: y <= y' & y' in X by A2,Def15;
consider z being Element of L such that
A5: z in X & x' <= z & y' <= z by A1,A3,A4;
take z; z <= z;
hence z in downarrow X by A5,Def15;
thus thesis by A3,A4,A5,ORDERS_1:26;
end;
set Y = downarrow X;
assume
A6: for x,y being Element of L st x in Y & y in Y
ex z being Element of L st z in Y & x <= z & y <= z;
let x,y be Element of L; assume
A7: x in X & y in X; x <= x & y <= y;
then x in Y & y in Y by A7,Def15;
then consider z being Element of L such that
A8: z in Y & x <= z & y <= z by A6;
consider z' being Element of L such that
A9: z <= z' & z' in X by A8,Def15;
take z'; thus z' in X by A9;
thus thesis by A8,A9,ORDERS_1:26;
end;
definition
let L be non empty transitive reflexive RelStr;
let X be directed Subset of L;
cluster downarrow X -> directed;
coherence by Th30;
end;
theorem Th31:
for L being non empty transitive reflexive RelStr, X be Subset of L
for x be Element of L holds x is_>=_than X iff x is_>=_than downarrow X
proof let L be non empty transitive reflexive RelStr, X be Subset of L;
let x be Element of L;
thus x is_>=_than X implies x is_>=_than downarrow X
proof assume
A1: for y being Element of L st y in X holds x >= y;
let y be Element of L; assume y in downarrow X;
then consider z being Element of L such that
A2: y <= z & z in X by Def15;
x >= z by A1,A2;
hence thesis by A2,ORDERS_1:26;
end;
assume
A3: for y being Element of L st y in downarrow X holds x >= y;
let y be Element of L; assume
A4: y in X; y <= y;
then y in downarrow X by A4,Def15;
hence thesis by A3;
end;
theorem Th32:
for L being non empty transitive reflexive RelStr, X be Subset of L holds
ex_sup_of X,L iff ex_sup_of downarrow X,L
proof let L be non empty transitive reflexive RelStr, X be Subset of L;
for x being Element of L holds x is_>=_than X iff x is_>=_than downarrow
X
by Th31;
hence thesis by YELLOW_0:46;
end;
theorem Th33:
for L being non empty transitive reflexive RelStr, X be Subset of L
st ex_sup_of X,L
holds sup X = sup downarrow X
proof let L be non empty transitive reflexive RelStr, X be Subset of L;
for x being Element of L holds x is_>=_than X iff x is_>=_than downarrow
X
by Th31;
hence thesis by YELLOW_0:47;
end;
theorem
for L being non empty Poset, x being Element of L holds
ex_sup_of downarrow x, L & sup downarrow x = x
proof let L be non empty Poset, x be Element of L;
A1: downarrow x = downarrow {x} & ex_sup_of {x}, L by Def17,YELLOW_0:38;
hence ex_sup_of downarrow x, L by Th32;
thus sup downarrow x = sup {x} by A1,Th33 .= x by YELLOW_0:39;
end;
theorem Th35:
for L being non empty transitive reflexive RelStr, X be Subset of L holds
X is filtered iff uparrow X is filtered
proof let L be non empty transitive reflexive RelStr, X be Subset of L;
thus X is filtered implies uparrow X is filtered
proof assume
A1: for x,y being Element of L st x in X & y in X
ex z being Element of L st z in X & x >= z & y >= z;
let x,y be Element of L; assume
A2: x in uparrow X & y in uparrow X;
then consider x' being Element of L such that
A3: x >= x' & x' in X by Def16;
consider y' being Element of L such that
A4: y >= y' & y' in X by A2,Def16;
consider z being Element of L such that
A5: z in X & x' >= z & y' >= z by A1,A3,A4;
take z; z >= z by ORDERS_1:24;
hence z in uparrow X by A5,Def16;
thus thesis by A3,A4,A5,ORDERS_1:26;
end;
set Y = uparrow X;
assume
A6: for x,y being Element of L st x in Y & y in Y
ex z being Element of L st z in Y & x >= z & y >= z;
let x,y be Element of L; assume
A7: x in X & y in X; x >= x & y >= y by ORDERS_1:24;
then x in Y & y in Y by A7,Def16;
then consider z being Element of L such that
A8: z in Y & x >= z & y >= z by A6;
consider z' being Element of L such that
A9: z >= z' & z' in X by A8,Def16;
take z'; thus z' in X by A9;
thus thesis by A8,A9,ORDERS_1:26;
end;
definition
let L be non empty transitive reflexive RelStr;
let X be filtered Subset of L;
cluster uparrow X -> filtered;
coherence by Th35;
end;
theorem Th36:
for L being non empty transitive reflexive RelStr, X be Subset of L
for x be Element of L holds
x is_<=_than X iff x is_<=_than uparrow X
proof let L be non empty transitive reflexive RelStr, X be Subset of L;
let x be Element of L;
thus x is_<=_than X implies x is_<=_than uparrow X
proof assume
A1: for y being Element of L st y in X holds x <= y;
let y be Element of L; assume y in uparrow X;
then consider z being Element of L such that
A2: y >= z & z in X by Def16;
x <= z by A1,A2;
hence thesis by A2,ORDERS_1:26;
end;
assume
A3: for y being Element of L st y in uparrow X holds x <= y;
let y be Element of L; assume
A4: y in X; y <= y;
then y in uparrow X by A4,Def16;
hence thesis by A3;
end;
theorem Th37:
for L being non empty transitive reflexive RelStr, X be Subset of L holds
ex_inf_of X,L iff ex_inf_of uparrow X,L
proof let L be non empty transitive reflexive RelStr, X be Subset of L;
for x being Element of L holds x is_<=_than X iff x is_<=_than uparrow X
by Th36;
hence thesis by YELLOW_0:48;
end;
theorem Th38:
for L being non empty transitive reflexive RelStr, X be Subset of L
st ex_inf_of X,L
holds inf X = inf uparrow X
proof let L be non empty transitive reflexive RelStr, X be Subset of L;
for x being Element of L holds x is_<=_than X iff x is_<=_than uparrow X
by Th36;
hence thesis by YELLOW_0:49;
end;
theorem
for L being non empty Poset, x being Element of L holds
ex_inf_of uparrow x, L & inf uparrow x = x
proof let L be non empty Poset, x be Element of L;
A1: uparrow x = uparrow {x} & ex_inf_of {x}, L by Def18,YELLOW_0:38;
hence ex_inf_of uparrow x, L by Th37;
thus inf uparrow x = inf {x} by A1,Th38 .= x by YELLOW_0:39;
end;
begin :: Ideals and filters
definition
let L be non empty reflexive transitive RelStr;
mode Ideal of L is directed lower non empty Subset of L;
:: Definition 1.3 (vii)
mode Filter of L is filtered upper non empty Subset of L;
:: Definition 1.3 (viii)
end;
theorem Th40:
for L being with_suprema antisymmetric RelStr, X being lower Subset of L holds
X is directed iff for x,y being Element of L st x in X & y in X holds x"\/"
y in
X
proof let L be with_suprema antisymmetric RelStr, X be lower Subset of L;
thus X is directed implies
for x,y being Element of L st x in X & y in X holds x"\/"y in X
proof assume
A1: for x,y being Element of L st x in X & y in X
ex z being Element of L st z in X & x <= z & y <= z;
let x,y be Element of L; assume x in X & y in X;
then consider z being Element of L such that
A2: z in X & x <= z & y <= z by A1;
x"\/"y <= z by A2,YELLOW_0:22;
hence x"\/"y in X by A2,Def19;
end;
assume
A3: for x,y being Element of L st x in X & y in X holds x"\/"y in X;
let x,y be Element of L; assume x in X & y in X;
then x"\/"y in X & x <= x"\/"y & y <= x"\/"y by A3,YELLOW_0:22;
hence thesis;
end;
theorem Th41:
for L being with_infima antisymmetric RelStr, X being upper Subset of L holds
X is filtered iff for x,y being Element of L st x in X & y in X holds x"/\"
y in
X
proof let L be with_infima antisymmetric RelStr, X be upper Subset of L;
thus X is filtered implies
for x,y being Element of L st x in X & y in X holds x"/\"y in X
proof assume
A1: for x,y being Element of L st x in X & y in X
ex z being Element of L st z in X & x >= z & y >= z;
let x,y be Element of L; assume x in X & y in X;
then consider z being Element of L such that
A2: z in X & x >= z & y >= z by A1;
x"/\"y >= z by A2,YELLOW_0:23;
hence x"/\"y in X by A2,Def20;
end;
assume
A3: for x,y being Element of L st x in X & y in X holds x"/\"y in X;
let x,y be Element of L; assume x in X & y in X;
then x"/\"y in X & x >= x"/\"y & y >= x"/\"y by A3,YELLOW_0:23;
hence thesis;
end;
theorem Th42:
for L being with_suprema Poset
for X being non empty lower Subset of L holds
X is directed iff for Y being finite Subset of X st Y <> {} holds "\/"
(Y,L) in
X
proof let L be with_suprema Poset;
let X be non empty lower Subset of L;
thus X is directed implies
for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in X
proof assume
A1: X is directed;
let Y be finite Subset of X such that
A2: Y <> {};
consider z being Element of L such that
A3: z in X & Y is_<=_than z by A1,Th1;
Y c= the carrier of L by XBOOLE_1:1;
then Y is finite Subset of L;
then ex_sup_of Y,L by A2,YELLOW_0:54;
then "\/"(Y,L) <= z by A3,YELLOW_0:30;
hence thesis by A3,Def19;
end;
assume
A4: for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in X;
consider x being Element of X;
reconsider x as Element of L;
now let Y be finite Subset of X;
per cases; suppose Y = {};
then x is_>=_than Y by YELLOW_0:6;
hence ex x being Element of L st x in X & x is_>=_than Y;
suppose
A5: Y <> {};
Y c= the carrier of L by XBOOLE_1:1;
then Y is finite Subset of L;
then ex_sup_of Y,L by A5,YELLOW_0:54;
then "\/"(Y,L) is_>=_than Y & "\/"(Y,L) in X by A4,A5,YELLOW_0:30;
hence ex x being Element of L st x in X & x is_>=_than Y;
end;
hence thesis by Th1;
end;
theorem Th43:
for L being with_infima Poset
for X being non empty upper Subset of L holds
X is filtered iff for Y being finite Subset of X st Y <> {} holds "/\"
(Y,L) in
X
proof let L be with_infima Poset, X be non empty upper Subset of L;
thus X is filtered implies
for Y being finite Subset of X st Y <> {} holds "/\"(Y,L) in X
proof assume
A1: X is filtered;
let Y be finite Subset of X such that
A2: Y <> {};
consider z being Element of L such that
A3: z in X & Y is_>=_than z by A1,Th2;
Y c= the carrier of L by XBOOLE_1:1;
then Y is finite Subset of L;
then ex_inf_of Y,L by A2,YELLOW_0:55;
then "/\"(Y,L) >= z by A3,YELLOW_0:31;
hence thesis by A3,Def20;
end;
assume
A4: for Y being finite Subset of X st Y <> {} holds "/\"(Y,L) in X;
consider x being Element of X;
reconsider x as Element of L;
now let Y be finite Subset of X;
per cases; suppose Y = {};
then x is_<=_than Y by YELLOW_0:6;
hence ex x being Element of L st x in X & x is_<=_than Y;
suppose
A5: Y <> {};
Y c= the carrier of L by XBOOLE_1:1;
then Y is finite Subset of L;
then ex_inf_of Y,L by A5,YELLOW_0:55;
then "/\"(Y,L) in X & "/\"(Y,L) is_<=_than Y by A4,A5,YELLOW_0:31;
hence ex x being Element of L st x in X & x is_<=_than Y;
end;
hence thesis by Th2;
end;
theorem
for L being non empty antisymmetric RelStr
st L is with_suprema or L is with_infima
for X,Y being Subset of L
st X is lower directed & Y is lower directed
holds X /\ Y is directed
proof let L be non empty antisymmetric RelStr such that
A1: L is with_suprema or L is with_infima;
let X,Y be Subset of L such that
A2: X is lower directed and
A3: Y is lower directed;
A4: X /\ Y is lower by A2,A3,Th27;
per cases by A1; suppose
A5: L is with_suprema;
now let x,y be Element of L; assume x in X /\ Y & y in X /\ Y;
then x in X & x in Y & y in X & y in Y by XBOOLE_0:def 3;
then x"\/"y in X & x"\/"y in Y by A2,A3,A5,Th40;
hence x"\/"y in X /\ Y by XBOOLE_0:def 3;
end;
hence thesis by A4,A5,Th40;
suppose
A6: L is with_infima;
let x,y be Element of L; assume x in X /\ Y & y in X /\ Y;
then A7: x in X & x in Y & y in X & y in Y by XBOOLE_0:def 3;
then consider zx being Element of L such that
A8: zx in X & x <= zx & y <= zx by A2,Def1;
consider zy being Element of L such that
A9: zy in Y & x <= zy & y <= zy by A3,A7,Def1;
take z = zx"/\"zy;
z <= zx & z <= zy by A6,YELLOW_0:23;
then z in X & z in Y by A2,A3,A8,A9,Def19;
hence z in X /\ Y by XBOOLE_0:def 3;
thus thesis by A6,A8,A9,YELLOW_0:23;
end;
theorem
for L being non empty antisymmetric RelStr
st L is with_suprema or L is with_infima
for X,Y being Subset of L
st X is upper filtered & Y is upper filtered
holds X /\ Y is filtered
proof let L be non empty antisymmetric RelStr such that
A1: L is with_suprema or L is with_infima;
let X,Y be Subset of L such that
A2: X is upper filtered and
A3: Y is upper filtered;
A4: X /\ Y is upper by A2,A3,Th29;
per cases by A1; suppose
A5: L is with_infima;
now let x,y be Element of L; assume x in X /\ Y & y in X /\ Y;
then x in X & x in Y & y in X & y in Y by XBOOLE_0:def 3;
then x"/\"y in X & x"/\"y in Y by A2,A3,A5,Th41;
hence x"/\"y in X /\ Y by XBOOLE_0:def 3;
end;
hence thesis by A4,A5,Th41;
suppose
A6: L is with_suprema;
let x,y be Element of L; assume x in X /\ Y & y in X /\ Y;
then A7: x in X & x in Y & y in X & y in Y by XBOOLE_0:def 3;
then consider zx being Element of L such that
A8: zx in X & x >= zx & y >= zx by A2,Def2;
consider zy being Element of L such that
A9: zy in Y & x >= zy & y >= zy by A3,A7,Def2;
take z = zx"\/"zy;
z >= zx & z >= zy by A6,YELLOW_0:22;
then z in X & z in Y by A2,A3,A8,A9,Def20;
hence z in X /\ Y by XBOOLE_0:def 3;
thus thesis by A6,A8,A9,YELLOW_0:22;
end;
theorem
for L being RelStr, A being Subset of bool the carrier of L st
(for X being Subset of L st X in A holds X is directed) &
(for X,Y being Subset of L st X in A & Y in A
ex Z being Subset of L st Z in A & X \/ Y c= Z)
for X being Subset of L st X = union A holds X is directed
proof let L be RelStr, A be Subset of bool the carrier of L such that
A1: for X being Subset of L st X in A holds X is directed and
A2: for X,Y being Subset of L st X in A & Y in A
ex Z being Subset of L st Z in A & X \/ Y c= Z;
let Z be Subset of L; assume
A3: Z = union A;
let x,y be Element of L; assume
x in Z;
then consider X being set such that
A4: x in X & X in A by A3,TARSKI:def 4;
assume y in Z;
then consider Y being set such that
A5: y in Y & Y in A by A3,TARSKI:def 4;
reconsider X,Y as Subset of L by A4,A5;
consider W being Subset of L such that
A6: W in A & X \/ Y c= W by A2,A4,A5;
X c= X \/ Y & Y c= X \/ Y by XBOOLE_1:7;
then x in X \/ Y & y in X \/ Y by A4,A5;
then x in W & y in W & W is directed by A1,A6;
then consider z being Element of L such that
A7: z in W & x <= z & y <= z by Def1;
take z; thus thesis by A3,A6,A7,TARSKI:def 4;
end;
theorem
for L being RelStr, A being Subset of bool the carrier of L st
(for X being Subset of L st X in A holds X is filtered) &
(for X,Y being Subset of L st X in A & Y in A
ex Z being Subset of L st Z in A & X \/ Y c= Z)
for X being Subset of L st X = union A holds X is filtered
proof let L be RelStr, A be Subset of bool the carrier of L such that
A1: for X being Subset of L st X in A holds X is filtered and
A2: for X,Y being Subset of L st X in A & Y in A
ex Z being Subset of L st Z in A & X \/ Y c= Z;
let Z be Subset of L; assume
A3: Z = union A;
let x,y be Element of L; assume
x in Z;
then consider X being set such that
A4: x in X & X in A by A3,TARSKI:def 4;
assume y in Z;
then consider Y being set such that
A5: y in Y & Y in A by A3,TARSKI:def 4;
reconsider X,Y as Subset of L by A4,A5;
consider W being Subset of L such that
A6: W in A & X \/ Y c= W by A2,A4,A5;
X c= X \/ Y & Y c= X \/ Y by XBOOLE_1:7;
then x in X \/ Y & y in X \/ Y by A4,A5;
then x in W & y in W & W is filtered by A1,A6;
then consider z being Element of L such that
A7: z in W & x >= z & y >= z by Def2;
take z; thus thesis by A3,A6,A7,TARSKI:def 4;
end;
definition
let L be non empty reflexive transitive RelStr;
let I be Ideal of L;
attr I is principal means
ex x being Element of L st x in I & x is_>=_than I;
end;
definition
let L be non empty reflexive transitive RelStr;
let F be Filter of L;
attr F is principal means
ex x being Element of L st x in F & x is_<=_than F;
end;
theorem
for L being non empty reflexive transitive RelStr, I being Ideal of L holds
I is principal iff ex x being Element of L st I = downarrow x
proof let L be non empty reflexive transitive RelStr, I be Ideal of L;
thus I is principal implies ex x being Element of L st I = downarrow x
proof given x being Element of L such that
A1: x in I & x is_>=_than I;
take x;
thus I c= downarrow x
proof let y be set; assume
A2: y in I;
then reconsider y as Element of L;
x >= y by A1,A2,LATTICE3:def 9;
hence thesis by Th17;
end;
let z be set; assume
A3: z in downarrow x;
then reconsider z as Element of L;
z <= x by A3,Th17;
hence thesis by A1,Def19;
end;
given x being Element of L such that
A4: I = downarrow x;
take x; x <= x;
hence x in I by A4,Th17;
let y be Element of L; thus thesis by A4,Th17;
end;
theorem
for L being non empty reflexive transitive RelStr, F being Filter of L holds
F is principal iff ex x being Element of L st F = uparrow x
proof let L be non empty reflexive transitive RelStr, I be Filter of L;
thus I is principal implies ex x being Element of L st I = uparrow x
proof given x being Element of L such that
A1: x in I & x is_<=_than I;
take x;
thus I c= uparrow x
proof let y be set; assume
A2: y in I;
then reconsider y as Element of L;
x <= y by A1,A2,LATTICE3:def 8;
hence thesis by Th18;
end;
let z be set; assume
A3: z in uparrow x;
then reconsider z as Element of L;
z >= x by A3,Th18;
hence thesis by A1,Def20;
end;
given x being Element of L such that
A4: I = uparrow x;
take x; x <= x;
hence x in I by A4,Th18;
let y be Element of L; thus thesis by A4,Th18;
end;
definition let L be non empty reflexive transitive RelStr;
func Ids L -> set equals :: Definition 1.3 (xi)
{X where X is Ideal of L: not contradiction};
correctness;
func Filt L -> set equals :: Definition 1.3 (xi)
{X where X is Filter of L: not contradiction};
correctness;
end;
definition let L be non empty reflexive transitive RelStr;
func Ids_0 L -> set equals :: Definition 1.3 (xii)
Ids L \/ {{}};
correctness;
func Filt_0 L -> set equals :: Definition 1.3 (xiii)
Filt L \/ {{}};
correctness;
end;
definition
let L be non empty RelStr;
let X be Subset of L;
set D = {"\/"(Y,L) where Y is finite Subset of X: ex_sup_of Y,L};
A1: D c= the carrier of L
proof let x be set; assume x in D;
then ex Y being finite Subset of X st x = "\/"(Y,L) & ex_sup_of Y,L;
hence thesis;
end;
func finsups X -> Subset of L equals:Def27:
{"\/"(Y,L) where Y is finite Subset of X: ex_sup_of Y,L};
correctness by A1;
set D = {"/\"(Y,L) where Y is finite Subset of X: ex_inf_of Y,L};
A2: D c= the carrier of L
proof let x be set; assume x in D;
then ex Y being finite Subset of X st x = "/\"(Y,L) & ex_inf_of Y,L;
hence thesis;
end;
func fininfs X -> Subset of L equals:Def28:
{"/\"(Y,L) where Y is finite Subset of X: ex_inf_of Y,L};
correctness by A2;
end;
definition
let L be non empty antisymmetric lower-bounded RelStr;
let X be Subset of L;
cluster finsups X -> non empty;
coherence
proof consider Z being empty Subset of X;
finsups X = {"\/"(Y,L) where Y is finite Subset of X: ex_sup_of Y,L} &
ex_sup_of {},L by Def27,YELLOW_0:42;
then "\/"(Z,L) in finsups X;
hence thesis;
end;
end;
definition
let L be non empty antisymmetric upper-bounded RelStr;
let X be Subset of L;
cluster fininfs X -> non empty;
coherence
proof consider Z being empty Subset of X;
fininfs X = {"/\"(Y,L) where Y is finite Subset of X: ex_inf_of Y,L} &
ex_inf_of {},L by Def28,YELLOW_0:43;
then "/\"(Z,L) in fininfs X;
hence thesis;
end;
end;
definition
let L be non empty reflexive antisymmetric RelStr;
let X be non empty Subset of L;
cluster finsups X -> non empty;
coherence
proof consider x being Element of X;
reconsider y = x as Element of L;
reconsider Z = {y} as finite Subset of X by ZFMISC_1:37;
finsups X = {"\/"(Y,L) where Y is finite Subset of X: ex_sup_of Y,L} &
ex_sup_of Z,L by Def27,YELLOW_0:38;
then "\/"(Z,L) in finsups X;
hence thesis;
end;
cluster fininfs X -> non empty;
coherence
proof consider x being Element of X;
reconsider y = x as Element of L;
reconsider Z = {y} as finite Subset of X by ZFMISC_1:37;
fininfs X = {"/\"(Y,L) where Y is finite Subset of X: ex_inf_of Y,L} &
ex_inf_of Z,L by Def28,YELLOW_0:38;
then "/\"(Z,L) in fininfs X;
hence thesis;
end;
end;
theorem Th50:
for L being non empty reflexive antisymmetric RelStr
for X being Subset of L
holds X c= finsups X & X c= fininfs X
proof let L be non empty reflexive antisymmetric RelStr;
let X be Subset of L;
A1: finsups X = {"\/"(Y,L) where Y is finite Subset of X: ex_sup_of Y,L}
by Def27;
hereby let x be set; assume
A2: x in X;
then reconsider y = x as Element of L;
reconsider Y = {x} as finite Subset of X by A2,ZFMISC_1:37;
y = "\/"(Y,L) & ex_sup_of {y},L by YELLOW_0:38,39;
hence x in finsups X by A1;
end;
A3: fininfs X = {"/\"(Y,L) where Y is finite Subset of X: ex_inf_of Y,L}
by Def28;
hereby let x be set; assume
A4: x in X;
then reconsider y = x as Element of L;
reconsider Y = {x} as finite Subset of X by A4,ZFMISC_1:37;
y = "/\"(Y,L) & ex_inf_of {y},L by YELLOW_0:38,39;
hence x in fininfs X by A3;
end;
end;
theorem Th51:
for L being non empty transitive RelStr
for X,F being Subset of L st
(for Y being finite Subset of X st Y <> {} holds ex_sup_of Y,L) &
(for x being Element of L st x in F
ex Y being finite Subset of X st ex_sup_of Y,L & x = "\/"(Y,L)) &
(for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in F)
holds F is directed
proof let L be non empty transitive RelStr;
let X,F be Subset of L such that
A1: for Y being finite Subset of X st Y <> {} holds ex_sup_of Y,L and
A2: for x being Element of L st x in F
ex Y being finite Subset of X st ex_sup_of Y,L & x = "\/"(Y,L) and
A3: for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in F;
let x,y be Element of L; assume
A4: x in F;
then consider Y1 being finite Subset of X such that
A5: ex_sup_of Y1,L & x = "\/"(Y1,L) by A2;
assume y in F;
then consider Y2 being finite Subset of X such that
A6: ex_sup_of Y2,L & y = "\/"(Y2,L) by A2;
take z = "\/"(Y1 \/ Y2, L);
A7: Y1 = {} & Y2 = {} & {} \/ {} = {} or Y1 \/ Y2 <> {} by XBOOLE_1:15;
hence z in F by A3,A4,A5;
ex_sup_of Y1 \/ Y2,L & Y1 c= Y1 \/ Y2 & Y2 c= Y1 \/ Y2 by A1,A5,A7,
XBOOLE_1:7;
hence thesis by A5,A6,YELLOW_0:34;
end;
definition
let L be with_suprema Poset;
let X be Subset of L;
cluster finsups X -> directed;
coherence
proof reconsider X as Subset of L;
A1: now let Y be finite Subset of X;
Y c= the carrier of L by XBOOLE_1:1;
then Y is Subset of L;
hence Y <> {} implies ex_sup_of Y,L by YELLOW_0:54;
end;
A2: finsups X = {"\/"(Y,L) where Y is finite Subset of X: ex_sup_of Y,L}
by Def27;
A3: now let x be Element of L; assume x in finsups X;
then ex Y being finite Subset of X st x = "\/"(Y,L) & ex_sup_of Y,L by A2
;
hence ex Y being finite Subset of X st ex_sup_of Y,L & x = "\/"(Y,L);
end;
now let Y be finite Subset of X;
Y c= the carrier of L by XBOOLE_1:1;
then reconsider Z = Y as Subset of L;
assume
Y <> {};
then ex_sup_of Z,L by YELLOW_0:54;
hence "\/"(Y,L) in finsups X by A2;
end;
hence thesis by A1,A3,Th51;
end;
end;
theorem Th52:
for L being non empty transitive reflexive RelStr, X,F being Subset of L st
(for Y being finite Subset of X st Y <> {} holds ex_sup_of Y,L) &
(for x being Element of L st x in F
ex Y being finite Subset of X st ex_sup_of Y,L & x = "\/"(Y,L)) &
(for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in F)
for x being Element of L holds x is_>=_than X iff x is_>=_than F
proof let L be non empty transitive reflexive RelStr;
let X,F be Subset of L such that
A1: for Y being finite Subset of X st Y <> {} holds ex_sup_of Y,L and
A2: for x being Element of L st x in F
ex Y being finite Subset of X st ex_sup_of Y,L & x = "\/"(Y,L) and
A3: for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in F;
let x be Element of L;
thus x is_>=_than X implies x is_>=_than F
proof assume
A4: x is_>=_than X;
let y be Element of L; assume y in F;
then consider Y being finite Subset of X such that
A5: ex_sup_of Y,L & y = "\/"(Y,L) by A2;
x is_>=_than Y by A4,YELLOW_0:9;
hence x >= y by A5,YELLOW_0:def 9;
end;
assume
A6: x is_>=_than F;
let y be Element of L; assume y in X;
then {y} c= X by ZFMISC_1:37;
then sup {y} in F & ex_sup_of {y},L by A1,A3;
then A7: {y} is_<=_than sup {y} & sup {y} <= x
by A6,LATTICE3:def 9,YELLOW_0:def 9;
then y <= sup {y} by YELLOW_0:7;
hence thesis by A7,ORDERS_1:26;
end;
theorem Th53:
for L being non empty transitive reflexive RelStr, X,F being Subset of L st
(for Y being finite Subset of X st Y <> {} holds ex_sup_of Y,L) &
(for x being Element of L st x in F
ex Y being finite Subset of X st ex_sup_of Y,L & x = "\/"(Y,L)) &
(for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in F)
holds ex_sup_of X,L iff ex_sup_of F,L
proof let L be non empty transitive reflexive RelStr;
let X,F be Subset of L such that
A1: for Y being finite Subset of X st Y <> {} holds ex_sup_of Y,L and
A2: for x being Element of L st x in F
ex Y being finite Subset of X st ex_sup_of Y,L & x = "\/"(Y,L) and
A3: for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in F;
for x being Element of L holds x is_>=_than X iff x is_>=_than F
by A1,A2,A3,Th52;
hence thesis by YELLOW_0:46;
end;
theorem Th54:
for L being non empty transitive reflexive RelStr, X,F being Subset of L st
(for Y being finite Subset of X st Y <> {} holds ex_sup_of Y,L) &
(for x being Element of L st x in F
ex Y being finite Subset of X st ex_sup_of Y,L & x = "\/"(Y,L)) &
(for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in F) &
ex_sup_of X,L
holds sup F = sup X
proof let L be non empty transitive reflexive RelStr;
let X,F be Subset of L such that
A1: for Y being finite Subset of X st Y <> {} holds ex_sup_of Y,L and
A2: for x being Element of L st x in F
ex Y being finite Subset of X st ex_sup_of Y,L & x = "\/"(Y,L) and
A3: for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in F;
for x being Element of L holds x is_>=_than X iff x is_>=_than F
by A1,A2,A3,Th52;
hence thesis by YELLOW_0:47;
end;
theorem
for L being with_suprema Poset, X being Subset of L
st ex_sup_of X,L or L is complete
holds sup X = sup finsups X
proof let L be with_suprema Poset, X be Subset of L; assume
ex_sup_of X,L or L is complete;
then A1: ex_sup_of X,L by YELLOW_0:17;
A2: now let Y be finite Subset of X;
Y c= the carrier of L by XBOOLE_1:1;
then Y is Subset of L;
hence Y <> {} implies ex_sup_of Y,L by YELLOW_0:54;
end;
A3: finsups X = {"\/"(Y,L) where Y is finite Subset of X: ex_sup_of Y,L}
by Def27;
A4: now let x be Element of L; assume x in finsups X;
then ex Y being finite Subset of X st x = "\/"(Y,L) & ex_sup_of Y,L by A3
;
hence ex Y being finite Subset of X st ex_sup_of Y,L & x = "\/"(Y,L);
end;
now let Y be finite Subset of X;
Y c= the carrier of L by XBOOLE_1:1;
then reconsider Z = Y as Subset of L;
assume
Y <> {};
then ex_sup_of Z,L by YELLOW_0:54;
hence "\/"(Y,L) in finsups X by A3;
end;
hence thesis by A1,A2,A4,Th54;
end;
theorem Th56:
for L being non empty transitive RelStr
for X,F being Subset of L st
(for Y being finite Subset of X st Y <> {} holds ex_inf_of Y,L) &
(for x being Element of L st x in F
ex Y being finite Subset of X st ex_inf_of Y,L & x = "/\"(Y,L)) &
(for Y being finite Subset of X st Y <> {} holds "/\"(Y,L) in F)
holds F is filtered
proof let L be non empty transitive RelStr;
let X,F be Subset of L such that
A1: for Y being finite Subset of X st Y <> {} holds ex_inf_of Y,L and
A2: for x being Element of L st x in F
ex Y being finite Subset of X st ex_inf_of Y,L & x = "/\"(Y,L) and
A3: for Y being finite Subset of X st Y <> {} holds "/\"(Y,L) in F;
let x,y be Element of L; assume
A4: x in F;
then consider Y1 being finite Subset of X such that
A5: ex_inf_of Y1,L & x = "/\"(Y1,L) by A2;
assume y in F;
then consider Y2 being finite Subset of X such that
A6: ex_inf_of Y2,L & y = "/\"(Y2,L) by A2;
take z = "/\"(Y1 \/ Y2, L);
A7: Y1 = {} & Y2 = {} & {} \/ {} = {} or Y1 \/ Y2 <> {} by XBOOLE_1:15;
hence z in F by A3,A4,A5;
ex_inf_of Y1 \/ Y2,L & Y1 c= Y1 \/ Y2 & Y2 c= Y1 \/ Y2 by A1,A5,A7,
XBOOLE_1:7;
hence thesis by A5,A6,YELLOW_0:35;
end;
definition
let L be with_infima Poset;
let X be Subset of L;
cluster fininfs X -> filtered;
coherence
proof reconsider X as Subset of L;
A1: fininfs X = {"/\"(Y,L) where Y is finite Subset of X: ex_inf_of Y,L}
by Def28;
A2: now let Y be finite Subset of X;
Y c= the carrier of L by XBOOLE_1:1;
then Y is Subset of L;
hence Y <> {} implies ex_inf_of Y,L by YELLOW_0:55;
end;
A3: now let x be Element of L; assume x in fininfs X;
then ex Y being finite Subset of X st x = "/\"(Y,L) & ex_inf_of Y,L by A1
;
hence ex Y being finite Subset of X st ex_inf_of Y,L & x = "/\"(Y,L);
end;
now let Y be finite Subset of X;
Y c= the carrier of L by XBOOLE_1:1;
then reconsider Z = Y as Subset of L;
assume
Y <> {};
then ex_inf_of Z,L by YELLOW_0:55;
hence "/\"(Y,L) in fininfs X by A1;
end;
hence thesis by A2,A3,Th56;
end;
end;
theorem Th57:
for L being non empty transitive reflexive RelStr, X,F being Subset of L st
(for Y being finite Subset of X st Y <> {} holds ex_inf_of Y,L) &
(for x being Element of L st x in F
ex Y being finite Subset of X st ex_inf_of Y,L & x = "/\"(Y,L)) &
(for Y being finite Subset of X st Y <> {} holds "/\"(Y,L) in F)
for x being Element of L holds x is_<=_than X iff x is_<=_than F
proof let L be non empty transitive reflexive RelStr;
let X,F be Subset of L such that
A1: for Y being finite Subset of X st Y <> {} holds ex_inf_of Y,L and
A2: for x being Element of L st x in F
ex Y being finite Subset of X st ex_inf_of Y,L & x = "/\"(Y,L) and
A3: for Y being finite Subset of X st Y <> {} holds "/\"(Y,L) in F;
let x be Element of L;
thus x is_<=_than X implies x is_<=_than F
proof assume
A4: x is_<=_than X;
let y be Element of L; assume y in F;
then consider Y being finite Subset of X such that
A5: ex_inf_of Y,L & y = "/\"(Y,L) by A2;
x is_<=_than Y by A4,YELLOW_0:9;
hence x <= y by A5,YELLOW_0:def 10;
end;
assume
A6: x is_<=_than F;
let y be Element of L; assume y in X;
then {y} c= X by ZFMISC_1:37;
then inf {y} in F & ex_inf_of {y},L by A1,A3;
then A7: {y} is_>=_than inf {y} & inf {y} >= x
by A6,LATTICE3:def 8,YELLOW_0:def 10;
then y >= inf {y} by YELLOW_0:7;
hence thesis by A7,ORDERS_1:26;
end;
theorem Th58:
for L being non empty transitive reflexive RelStr, X,F being Subset of L st
(for Y being finite Subset of X st Y <> {} holds ex_inf_of Y,L) &
(for x being Element of L st x in F
ex Y being finite Subset of X st ex_inf_of Y,L & x = "/\"(Y,L)) &
(for Y being finite Subset of X st Y <> {} holds "/\"(Y,L) in F)
holds ex_inf_of X,L iff ex_inf_of F,L
proof let L be non empty transitive reflexive RelStr;
let X,F be Subset of L such that
A1: for Y being finite Subset of X st Y <> {} holds ex_inf_of Y,L and
A2: for x being Element of L st x in F
ex Y being finite Subset of X st ex_inf_of Y,L & x = "/\"(Y,L) and
A3: for Y being finite Subset of X st Y <> {} holds "/\"(Y,L) in F;
for x being Element of L holds x is_<=_than X iff x is_<=_than F
by A1,A2,A3,Th57;
hence thesis by YELLOW_0:48;
end;
theorem Th59:
for L being non empty transitive reflexive RelStr, X,F being Subset of L st
(for Y being finite Subset of X st Y <> {} holds ex_inf_of Y,L) &
(for x being Element of L st x in F
ex Y being finite Subset of X st ex_inf_of Y,L & x = "/\"(Y,L)) &
(for Y being finite Subset of X st Y <> {} holds "/\"(Y,L) in F) &
ex_inf_of X, L
holds inf F = inf X
proof let L be non empty transitive reflexive RelStr;
let X,F be Subset of L such that
A1: for Y being finite Subset of X st Y <> {} holds ex_inf_of Y,L and
A2: for x being Element of L st x in F
ex Y being finite Subset of X st ex_inf_of Y,L & x = "/\"(Y,L) and
A3: for Y being finite Subset of X st Y <> {} holds "/\"(Y,L) in F;
for x being Element of L holds x is_<=_than X iff x is_<=_than F
by A1,A2,A3,Th57;
hence thesis by YELLOW_0:49;
end;
theorem
for L being with_infima Poset, X being Subset of L
st ex_inf_of X,L or L is complete
holds inf X = inf fininfs X
proof let L be with_infima Poset, X be Subset of L; assume
ex_inf_of X,L or L is complete;
then A1: ex_inf_of X,L by YELLOW_0:17;
A2: now let Y be finite Subset of X;
Y c= the carrier of L by XBOOLE_1:1;
then Y is Subset of L;
hence Y <> {} implies ex_inf_of Y,L by YELLOW_0:55;
end;
A3: fininfs X = {"/\"(Y,L) where Y is finite Subset of X: ex_inf_of Y,L}
by Def28;
A4: now let x be Element of L; assume x in fininfs X;
then ex Y being finite Subset of X st x = "/\"(Y,L) & ex_inf_of Y,L by A3
;
hence ex Y being finite Subset of X st ex_inf_of Y,L & x = "/\"(Y,L);
end;
now let Y be finite Subset of X;
Y c= the carrier of L by XBOOLE_1:1;
then reconsider Z = Y as Subset of L;
assume
Y <> {};
then ex_inf_of Z,L by YELLOW_0:55;
hence "/\"(Y,L) in fininfs X by A3;
end;
hence thesis by A1,A2,A4,Th59;
end;
theorem
for L being with_suprema Poset, X being Subset of L holds
X c= downarrow finsups X &
for I being Ideal of L st X c= I holds downarrow finsups X c= I
proof let L be with_suprema Poset, X be Subset of L;
X c= finsups X & finsups X c= downarrow finsups X by Th16,Th50;
hence X c= downarrow finsups X by XBOOLE_1:1;
let I be Ideal of L such that
A1: X c= I;
let x be set; assume
A2: x in downarrow finsups X;
then reconsider x as Element of L;
consider y being Element of L such that
A3: x <= y & y in finsups X by A2,Def15;
finsups X = {"\/"(Y,L) where Y is finite Subset of X: ex_sup_of Y,L}
by Def27;
then consider Y being finite Subset of X such that
A4: y = "\/"(Y,L) & ex_sup_of Y,L by A3;
consider i being Element of I;
reconsider i as Element of L;
A5: ex_sup_of {i}, L & sup {i} = i & {} c= {i} by XBOOLE_1:2,YELLOW_0:38,39;
A6: now assume ex_sup_of {},L;
then "\/"({},L) <= sup {i} by A5,YELLOW_0:34;
hence "\/"({},L) in I by A5,Def19;
end;
Y c= I & (Y = {} or Y <> {}) by A1,XBOOLE_1:1;
then y in I by A4,A6,Th42;
hence thesis by A3,Def19;
end;
theorem
for L being with_infima Poset, X being Subset of L holds
X c= uparrow fininfs X &
for F being Filter of L st X c= F holds uparrow fininfs X c= F
proof let L be with_infima Poset, X be Subset of L;
X c= fininfs X & fininfs X c= uparrow fininfs X by Th16,Th50;
hence X c= uparrow fininfs X by XBOOLE_1:1;
let I be Filter of L such that
A1: X c= I;
let x be set; assume
A2: x in uparrow fininfs X;
then reconsider x as Element of L;
consider y being Element of L such that
A3: x >= y & y in fininfs X by A2,Def16;
fininfs X = {"/\"(Y,L) where Y is finite Subset of X: ex_inf_of Y,L}
by Def28;
then consider Y being finite Subset of X such that
A4: y = "/\"(Y,L) & ex_inf_of Y,L by A3;
consider i being Element of I;
reconsider i as Element of L;
A5: ex_inf_of {i}, L & inf {i} = i & {} c= {i} by XBOOLE_1:2,YELLOW_0:38,39;
A6: now assume ex_inf_of {},L;
then "/\"({},L) >= inf {i} by A5,YELLOW_0:35;
hence "/\"({},L) in I by A5,Def20;
end;
Y c= I & (Y = {} or Y <> {}) by A1,XBOOLE_1:1;
then y in I by A4,A6,Th43;
hence thesis by A3,Def20;
end;
begin :: Chains
definition
let L be non empty RelStr;
attr L is connected means:Def29:
for x,y being Element of L holds x <= y or y <= x;
end;
definition
cluster trivial -> connected (non empty reflexive RelStr);
coherence
proof let L be non empty reflexive RelStr; assume
A1: for x,y being Element of L holds x = y;
let z1,z2 be Element of L;
z1 = z2 by A1;
hence thesis by ORDERS_1:24;
end;
end;
definition
cluster connected (non empty Poset);
existence
proof consider O being Order of {0};
take L = RelStr(#{0},O#);
let x,y be Element of L; x = 0 & y = 0 by TARSKI:def 1;
hence thesis by ORDERS_1:24;
end;
end;
definition
mode Chain is connected (non empty Poset);
end;
definition
let L be Chain;
cluster L~ -> connected;
coherence
proof let x,y be Element of L~;
A1: ~x = x & ~y = y & (~x)~ = ~x & (~y)~ = ~y by LATTICE3:def 6,def 7;
~x <= ~y or ~x >= ~y by Def29;
hence thesis by A1,LATTICE3:9;
end;
end;
begin :: Semilattices
definition
mode Semilattice is with_infima Poset;
mode sup-Semilattice is with_suprema Poset;
mode LATTICE is with_infima with_suprema Poset;
end;
theorem
for L being Semilattice for X being upper non empty Subset of L holds
X is Filter of L iff subrelstr X is meet-inheriting
proof let L be Semilattice, X be upper non empty Subset of L;
set S = subrelstr X;
A1: the carrier of S = X by YELLOW_0:def 15;
hereby assume
A2: X is Filter of L;
thus S is meet-inheriting
proof let x,y be Element of L; assume
A3: x in the carrier of S & y in the carrier of S & ex_inf_of {x,y},L;
then consider z being Element of L such that
A4: z in X & x >= z & y >= z by A1,A2,Def2;
z is_<=_than {x,y} by A4,YELLOW_0:8;
then z <= inf {x,y} by A3,YELLOW_0:def 10;
hence inf {x,y} in the carrier of S by A1,A4,Def20;
end;
end;
assume
A5: for x,y being Element of L st
x in the carrier of S & y in the carrier of S & ex_inf_of {x,y},L
holds inf {x,y} in the carrier of S;
X is filtered
proof let x,y be Element of L; assume
A6: x in X & y in X;
take z = inf {x,y};
z = x "/\" y & ex_inf_of {x,y},L by YELLOW_0:21,40;
hence z in X & z <= x & z <= y by A1,A5,A6,YELLOW_0:19;
end;
hence X is Filter of L;
end;
theorem
for L being sup-Semilattice for X being lower non empty Subset of L holds
X is Ideal of L iff subrelstr X is join-inheriting
proof let L be sup-Semilattice, X be lower non empty Subset of L;
set S = subrelstr X;
A1: the carrier of S = X by YELLOW_0:def 15;
hereby assume
A2: X is Ideal of L;
thus S is join-inheriting
proof let x,y be Element of L; assume
A3: x in the carrier of S & y in the carrier of S & ex_sup_of {x,y},L;
then consider z being Element of L such that
A4: z in X & x <= z & y <= z by A1,A2,Def1;
z is_>=_than {x,y} by A4,YELLOW_0:8;
then z >= sup {x,y} by A3,YELLOW_0:def 9;
hence sup {x,y} in the carrier of S by A1,A4,Def19;
end;
end;
assume
A5: for x,y being Element of L st
x in the carrier of S & y in the carrier of S & ex_sup_of {x,y},L
holds sup {x,y} in the carrier of S;
X is directed
proof let x,y be Element of L; assume
A6: x in X & y in X;
take z = sup {x,y};
z = x "\/" y & ex_sup_of {x,y},L by YELLOW_0:20,41;
hence z in X & x <= z & y <= z by A1,A5,A6,YELLOW_0:18;
end;
hence X is Ideal of L;
end;
begin :: Maps
definition
let S,T be non empty RelStr;
let f be map of S,T;
let X be Subset of S;
pred f preserves_inf_of X means:Def30:
ex_inf_of X,S implies ex_inf_of f.:X,T & inf (f.:X) = f.inf X;
pred f preserves_sup_of X means:Def31:
ex_sup_of X,S implies ex_sup_of f.:X,T & sup (f.:X) = f.sup X;
end;
theorem
for S1,S2, T1,T2 being non empty RelStr st
the RelStr of S1 = the RelStr of T1 & the RelStr of S2 = the RelStr of T2
for f being map of S1,S2, g being map of T1,T2 st f = g
for X being Subset of S1, Y being Subset of T1 st X = Y holds
(f preserves_sup_of X implies g preserves_sup_of Y) &
(f preserves_inf_of X implies g preserves_inf_of Y)
proof let S1,S2, T1,T2 be non empty RelStr such that
A1: the RelStr of S1 = the RelStr of T1 & the RelStr of S2 = the RelStr of T2;
let f be map of S1,S2, g be map of T1,T2 such that
A2: f = g;
let X be Subset of S1, Y be Subset of T1 such that
A3: X = Y;
thus f preserves_sup_of X implies g preserves_sup_of Y
proof assume
A4: ex_sup_of X,S1 implies ex_sup_of f.:X,S2 & sup (f.:X) = f.sup X;
assume
A5: ex_sup_of Y,T1;
hence ex_sup_of g.:Y,T2 by A1,A2,A3,A4,YELLOW_0:14;
sup (f.:X) = sup (g.:
Y) & sup X = sup Y by A1,A2,A3,A4,A5,YELLOW_0:14,26;
hence thesis by A1,A2,A3,A4,A5,YELLOW_0:14;
end;
assume
A6: ex_inf_of X,S1 implies ex_inf_of f.:X,S2 & inf (f.:X) = f.inf X;
assume
A7: ex_inf_of Y,T1;
hence ex_inf_of g.:Y,T2 by A1,A2,A3,A6,YELLOW_0:14;
inf (f.:X) = inf (g.:Y) & inf X = inf Y by A1,A2,A3,A6,A7,YELLOW_0:14,27;
hence thesis by A1,A2,A3,A6,A7,YELLOW_0:14;
end;
definition
let L1,L2 be non empty RelStr;
let f be map of L1,L2;
attr f is infs-preserving means
for X being Subset of L1 holds f preserves_inf_of X;
attr f is sups-preserving means
for X being Subset of L1 holds f preserves_sup_of X;
attr f is meet-preserving means
for x,y being Element of L1 holds f preserves_inf_of {x,y};
attr f is join-preserving means
for x,y being Element of L1 holds f preserves_sup_of {x,y};
attr f is filtered-infs-preserving means
for X being Subset of L1 st X is non empty filtered
holds f preserves_inf_of X;
attr f is directed-sups-preserving means
for X being Subset of L1 st X is non empty directed
holds f preserves_sup_of X;
end;
definition
let L1,L2 be non empty RelStr;
cluster infs-preserving ->
filtered-infs-preserving meet-preserving map of L1,L2;
coherence
proof let f be map of L1,L2; assume
A1: for X being Subset of L1 holds f preserves_inf_of X;
hence for X being Subset of L1 st X is non empty filtered
holds f preserves_inf_of X;
thus for x,y being Element of L1 holds f preserves_inf_of {x,y} by A1;
end;
cluster sups-preserving ->
directed-sups-preserving join-preserving map of L1,L2;
coherence
proof let f be map of L1,L2; assume
A2: for X being Subset of L1 holds f preserves_sup_of X;
hence for X being Subset of L1 st X is non empty directed
holds f preserves_sup_of X;
thus for x,y being Element of L1 holds f preserves_sup_of {x,y} by A2;
end;
end;
definition
let S,T be RelStr;
let f be map of S,T;
attr f is isomorphic means:Def38:
f is one-to-one monotone & ex g being map of T,S st g = f" & g is monotone
if S is non empty & T is non empty
otherwise S is empty & T is empty;
correctness;
end;
theorem Th66:
for S,T being non empty RelStr, f being map of S,T holds f is isomorphic iff
f is one-to-one & rng f = the carrier of T &
for x,y being Element of S holds x <= y iff f.x <= f.y
proof let S,T be non empty RelStr; let f be map of S,T;
hereby assume
A1: f is isomorphic;
hence f is one-to-one by Def38;
consider g being map of T,S such that
A2: g = f" & g is monotone by A1,Def38;
A3: f is one-to-one monotone by A1,Def38;
then rng f = dom g & g is Function of the carrier of T, the carrier of S
by A2,FUNCT_1:55;
hence rng f = the carrier of T by FUNCT_2:def 1;
let x,y be Element of S;
thus x <= y implies f.x <= f.y by A3,ORDERS_3:def 5;
assume
A4: f.x <= f.y;
g.(f.x) = x & g.(f.y) = y by A2,A3,FUNCT_2:32;
hence x <= y by A2,A4,ORDERS_3:def 5;
end;
assume
A5: f is one-to-one & rng f = the carrier of T &
for x,y being Element of S holds x <= y iff f.x <= f.y;
per cases; case S is non empty & T is non empty;
thus f is one-to-one by A5;
thus for x,y being Element of S st x <= y
for a,b being Element of T st a = f.x & b = f.y holds a <= b by A5;
f" is Function of the carrier of T, the carrier of S by A5,FUNCT_2:31;
then reconsider g = f" as map of T,S;
take g; thus g = f";
let x,y be Element of T;
consider a being set such that
A6: a in dom f & x = f.a by A5,FUNCT_1:def 5;
consider b being set such that
A7: b in dom f & y = f.b by A5,FUNCT_1:def 5;
reconsider a,b as Element of S by A6,A7;
g.x = a & g.y = b by A5,A6,A7,FUNCT_2:32;
hence thesis by A5,A6,A7;
case S is empty or T is empty; hence thesis;
end;
definition
let S,T be non empty RelStr;
cluster isomorphic -> one-to-one monotone map of S,T;
coherence by Def38;
end;
theorem
for S,T being non empty RelStr, f being map of S,T st f is isomorphic
holds f" is map of T,S & rng (f") = the carrier of S
proof let S,T be non empty RelStr, f be map of S,T; assume
f is isomorphic;
then A1: f is one-to-one & rng f = the carrier of T & dom f = the carrier of S
by Th66,FUNCT_2:def 1;
then dom (f") = the carrier of T & rng (f") = the carrier of S
by FUNCT_1:55;
then f" is Function of the carrier of T, the carrier of S by FUNCT_2:3;
hence thesis by A1,FUNCT_1:55;
end;
theorem
for S,T being non empty RelStr, f being map of S,T st f is isomorphic
for g being map of T,S st g = f" holds g is isomorphic
proof let S,T be non empty RelStr, f be map of S,T; assume
A1: f is isomorphic;
then A2: f is one-to-one monotone by Def38;
consider h being map of T,S such that
A3: h = f" & h is monotone by A1,Def38;
let g be map of T,S; assume
A4: g = f";
per cases;
case T is non empty & S is non empty;
thus g is one-to-one monotone by A2,A3,A4,FUNCT_1:62;
f" " = f by A2,FUNCT_1:65;
hence thesis by A2,A4;
case T is empty or S is empty;
hence thesis;
end;
theorem Th69:
for S,T being non empty Poset, f being map of S,T st
for X being Filter of S holds f preserves_inf_of X
holds f is monotone
proof let S,T be non empty Poset, f be map of S,T; assume
A1: for X being Filter of S holds f preserves_inf_of X;
let x,y be Element of S;
A2: uparrow x = uparrow {x} & uparrow y = uparrow {y} by Def18;
A3: ex_inf_of {x}, S & ex_inf_of {y}, S by YELLOW_0:38;
then f preserves_inf_of uparrow x & f preserves_inf_of uparrow y &
ex_inf_of uparrow x, S & ex_inf_of uparrow y, S by A1,A2,Th37;
then A4: ex_inf_of f.:uparrow x, T & ex_inf_of f.:uparrow y, T &
inf (f.:uparrow x) = f.inf uparrow x & inf (f.:uparrow y) = f.inf uparrow y
by Def30;
assume x <= y;
then uparrow y c= uparrow x by Th22;
then A5: f.:uparrow y c= f.:uparrow x by RELAT_1:156;
inf (f.:uparrow x) = f.inf {x} & inf (f.:uparrow y) = f.inf {y}
by A2,A3,A4,Th38;
then inf (f.:uparrow x) = f.x & inf (f.:uparrow y) = f.y by YELLOW_0:39;
hence thesis by A4,A5,YELLOW_0:35;
end;
theorem
for S,T being non empty Poset, f being map of S,T st
for X being Filter of S holds f preserves_inf_of X
holds f is filtered-infs-preserving
proof let S,T be non empty Poset, f be map of S,T such that
A1: for X being Filter of S holds f preserves_inf_of X;
let X be Subset of S such that
A2: X is non empty filtered and
A3: ex_inf_of X,S;
reconsider Y = X as non empty filtered Subset of S by A2;
uparrow Y is Filter of S;
then f preserves_inf_of uparrow X & ex_inf_of uparrow X, S by A1,A3,Th37;
then A4: ex_inf_of f.:uparrow X,T & inf (f.:uparrow X) = f.inf uparrow X &
inf uparrow X = inf X by A3,Def30,Th38;
X c= uparrow X by Th16;
then A5: f.:X c= f.:uparrow X by RELAT_1:156;
A6: f.:uparrow X is_>=_than f.inf X by A4,YELLOW_0:31;
A7: f.:X is_>=_than f.inf X
proof let x be Element of T; assume x in f.:X;
hence thesis by A5,A6,LATTICE3:def 8;
end;
A8: now let b be Element of T; assume
A9: f.:X is_>=_than b;
f.:uparrow X is_>=_than b
proof let a be Element of T; assume a in f.:uparrow X;
then consider x being set such that
A10: x in dom f & x in uparrow X & a = f.x by FUNCT_1:def 12;
uparrow X =
{z where z is Element of S: ex y being Element of S st z >= y & y in
X}
by Th15;
then consider z being Element of S such that
A11: x = z & ex y being Element of S st z >= y & y in X by A10;
consider y being Element of S such that
A12: z >= y & y in X by A11;
f is monotone & f.y in f.:X by A1,A12,Th69,FUNCT_2:43;
then f.z >= f.y & f.y >= b by A9,A12,LATTICE3:def 8,ORDERS_3:def 5;
hence thesis by A10,A11,ORDERS_1:26;
end;
hence f.inf X >= b by A4,YELLOW_0:31;
end;
hence ex_inf_of f.:X,T by A7,YELLOW_0:16;
hence inf (f.:X) = f.inf X by A7,A8,YELLOW_0:def 10;
end;
theorem
for S being Semilattice, T being non empty Poset, f being map of S,T st
(for X being finite Subset of S holds f preserves_inf_of X) &
(for X being non empty filtered Subset of S holds f preserves_inf_of X)
holds f is infs-preserving
proof let S be Semilattice, T be non empty Poset, f be map of S,T such that
A1: for X being finite Subset of S holds f preserves_inf_of X and
A2: for X being non empty filtered Subset of S holds f preserves_inf_of X;
let X be Subset of S such that
A3: ex_inf_of X,S;
defpred P[set] means
ex Y being finite Subset of X st ex_inf_of Y, S & $1 = "/\"(Y,S);
consider Z being set such that
A4: for x being set holds x in Z iff x in the carrier of S & P[x]
from Separation;
Z c= the carrier of S proof let x be set; thus thesis by A4; end;
then reconsider Z as Subset of S;
A5: now let Y be finite Subset of X;
Y is Subset of S by XBOOLE_1:1;
then Y is finite Subset of S;
hence Y <> {} implies ex_inf_of Y, S by YELLOW_0:55;
end;
A6: now let Y be finite Subset of X;
Y is Subset of S by XBOOLE_1:1;
then reconsider Y' = Y as Subset of S;
assume Y <> {};
then ex_inf_of Y', S by YELLOW_0:55;
hence "/\"(Y,S) in Z by A4;
end;
for x being Element of S st x in Z
ex Y being finite Subset of X st ex_inf_of Y,S & x = "/\"(Y,S) by A4;
then A7: Z is filtered & ex_inf_of Z, S & inf Z = inf X & (Z = {} or Z <> {})
by A3,A5,A6,Th56,Th58,Th59;
then f preserves_inf_of Z by A1,A2;
then A8: ex_inf_of f.:Z,T & inf (f.:Z) = f.inf Z & inf Z = inf X by A7,Def30;
X c= Z
proof let x be set; assume
A9: x in X;
then reconsider Y = {x} as finite Subset of X by ZFMISC_1:37;
reconsider x as Element of S by A9;
Y is Subset of S by XBOOLE_1:1;
then Y is Subset of S;
then ex_inf_of Y, S & x = "/\"(Y,S) by YELLOW_0:39,55;
hence thesis by A4;
end;
then A10: f.:X c= f.:Z by RELAT_1:156;
A11: f.:Z is_>=_than f.inf X by A8,YELLOW_0:31;
A12: f.:X is_>=_than f.inf X
proof let x be Element of T; assume x in f.:X;
hence thesis by A10,A11,LATTICE3:def 8;
end;
A13: now let b be Element of T; assume
A14: f.:X is_>=_than b;
f.:Z is_>=_than b
proof let a be Element of T; assume a in f.:Z;
then consider x being set such that
A15: x in dom f & x in Z & a = f.x by FUNCT_1:def 12;
consider Y being finite Subset of X such that
A16: ex_inf_of Y, S & x = "/\"(Y,S) by A4,A15;
Y is Subset of S by XBOOLE_1:1;
then reconsider Y as Subset of S;
f.:Y c= f.:X & f preserves_inf_of Y by A1,RELAT_1:156;
then b is_<=_than f.:Y & a = "/\"(f.:Y,T) & ex_inf_of f.:Y, T
by A14,A15,A16,Def30,YELLOW_0:9;
hence thesis by YELLOW_0:def 10;
end;
hence f.inf X >= b by A8,YELLOW_0:31;
end;
hence ex_inf_of f.:X,T by A12,YELLOW_0:16;
hence inf (f.:X) = f.inf X by A12,A13,YELLOW_0:def 10;
end;
theorem Th72:
for S,T being non empty Poset, f being map of S,T st
for X being Ideal of S holds f preserves_sup_of X
holds f is monotone
proof let S,T be non empty Poset, f be map of S,T; assume
A1: for X being Ideal of S holds f preserves_sup_of X;
let x,y be Element of S;
A2: downarrow x = downarrow {x} & downarrow y = downarrow {y} by Def17;
A3: ex_sup_of {x}, S & ex_sup_of {y}, S by YELLOW_0:38;
then f preserves_sup_of downarrow x & f preserves_sup_of downarrow y &
ex_sup_of downarrow x, S & ex_sup_of downarrow y, S by A1,A2,Th32;
then A4: ex_sup_of f.:downarrow x, T & ex_sup_of f.:downarrow y, T &
sup (f.:downarrow x) = f.sup downarrow x &
sup (f.:downarrow y) = f.sup downarrow y by Def31;
assume x <= y;
then downarrow x c= downarrow y by Th21;
then A5: f.:downarrow x c= f.:downarrow y by RELAT_1:156;
sup (f.:downarrow x) = f.sup {x} & sup (f.:downarrow y) = f.sup {y}
by A2,A3,A4,Th33;
then sup (f.:downarrow x) = f.x & sup (f.:downarrow y) = f.y by YELLOW_0:39
;
hence thesis by A4,A5,YELLOW_0:34;
end;
theorem
for S,T being non empty Poset, f being map of S,T st
for X being Ideal of S holds f preserves_sup_of X
holds f is directed-sups-preserving
proof let S,T be non empty Poset, f be map of S,T such that
A1: for X being Ideal of S holds f preserves_sup_of X;
let X be Subset of S such that
A2: X is non empty directed and
A3: ex_sup_of X,S;
reconsider Y = X as non empty directed Subset of S by A2;
downarrow Y is Ideal of S;
then f preserves_sup_of downarrow X & ex_sup_of downarrow X, S by A1,A3,
Th32
;
then A4: ex_sup_of f.:downarrow X,T & sup (f.:downarrow X) = f.sup downarrow X
&
sup downarrow X = sup X by A3,Def31,Th33;
X c= downarrow X by Th16;
then A5: f.:X c= f.:downarrow X by RELAT_1:156;
A6: f.:downarrow X is_<=_than f.sup X by A4,YELLOW_0:30;
A7: f.:X is_<=_than f.sup X
proof let x be Element of T; assume x in f.:X;
hence thesis by A5,A6,LATTICE3:def 9;
end;
A8: now let b be Element of T; assume
A9: f.:X is_<=_than b;
f.:downarrow X is_<=_than b
proof let a be Element of T; assume a in f.:downarrow X;
then consider x being set such that
A10: x in dom f & x in downarrow X & a = f.x by FUNCT_1:def 12;
downarrow X =
{z where z is Element of S: ex y being Element of S st z <= y & y in
X}
by Th14;
then consider z being Element of S such that
A11: x = z & ex y being Element of S st z <= y & y in X by A10;
consider y being Element of S such that
A12: z <= y & y in X by A11;
f is monotone & f.y in f.:X by A1,A12,Th72,FUNCT_2:43;
then f.z <= f.y & f.y <= b by A9,A12,LATTICE3:def 9,ORDERS_3:def 5;
hence thesis by A10,A11,ORDERS_1:26;
end;
hence f.sup X <= b by A4,YELLOW_0:30;
end;
hence ex_sup_of f.:X,T by A7,YELLOW_0:15;
hence sup (f.:X) = f.sup X by A7,A8,YELLOW_0:def 9;
end;
theorem
for S being sup-Semilattice, T being non empty Poset, f being map of S,T st
(for X being finite Subset of S holds f preserves_sup_of X) &
(for X being non empty directed Subset of S holds f preserves_sup_of X)
holds f is sups-preserving
proof let S be sup-Semilattice, T be non empty Poset, f be map of S,T such
that
A1: for X being finite Subset of S holds f preserves_sup_of X and
A2: for X being non empty directed Subset of S holds f preserves_sup_of X;
let X be Subset of S such that
A3: ex_sup_of X,S;
defpred P[set] means
ex Y being finite Subset of X st ex_sup_of Y, S & $1 = "\/"(Y,S);
consider Z being set such that
A4: for x being set holds x in Z iff x in the carrier of S & P[x]
from Separation;
Z c= the carrier of S proof let x be set; thus thesis by A4; end;
then reconsider Z as Subset of S;
A5: now let Y be finite Subset of X;
Y is Subset of S by XBOOLE_1:1;
then Y is finite Subset of S;
hence Y <> {} implies ex_sup_of Y, S by YELLOW_0:54;
end;
A6: now let Y be finite Subset of X;
Y is Subset of S by XBOOLE_1:1;
then reconsider Y' = Y as Subset of S;
assume Y <> {};
then ex_sup_of Y', S by YELLOW_0:54;
hence "\/"(Y,S) in Z by A4;
end;
for x being Element of S st x in Z
ex Y being finite Subset of X st ex_sup_of Y,S & x = "\/"(Y,S) by A4;
then A7: Z is directed & ex_sup_of Z, S & sup Z = sup X & (Z = {} or Z <> {})
by A3,A5,A6,Th51,Th53,Th54;
then f preserves_sup_of Z by A1,A2;
then A8: ex_sup_of f.:Z,T & sup (f.:Z) = f.sup Z & sup Z = sup X by A7,Def31;
X c= Z
proof let x be set; assume
A9: x in X;
then reconsider Y = {x} as finite Subset of X by ZFMISC_1:37;
reconsider x as Element of S by A9;
Y is Subset of S by XBOOLE_1:1;
then Y is Subset of S;
then ex_sup_of Y, S & x = "\/"(Y,S) by YELLOW_0:39,54;
hence thesis by A4;
end;
then A10: f.:X c= f.:Z by RELAT_1:156;
A11: f.:Z is_<=_than f.sup X by A8,YELLOW_0:30;
A12: f.:X is_<=_than f.sup X
proof let x be Element of T; assume x in f.:X;
hence thesis by A10,A11,LATTICE3:def 9;
end;
A13: now let b be Element of T; assume
A14: f.:X is_<=_than b;
f.:Z is_<=_than b
proof let a be Element of T; assume a in f.:Z;
then consider x being set such that
A15: x in dom f & x in Z & a = f.x by FUNCT_1:def 12;
consider Y being finite Subset of X such that
A16: ex_sup_of Y, S & x = "\/"(Y,S) by A4,A15;
Y is Subset of S by XBOOLE_1:1;
then reconsider Y as Subset of S;
f.:Y c= f.:X & f preserves_sup_of Y by A1,RELAT_1:156;
then b is_>=_than f.:Y & a = "\/"(f.:Y,T) & ex_sup_of f.:Y, T
by A14,A15,A16,Def31,YELLOW_0:9;
hence thesis by YELLOW_0:def 9;
end;
hence f.sup X <= b by A8,YELLOW_0:30;
end;
hence ex_sup_of f.:X,T by A12,YELLOW_0:15;
hence sup (f.:X) = f.sup X by A12,A13,YELLOW_0:def 9;
end;
begin :: Complete Semilattice
definition
let L be non empty reflexive RelStr;
attr L is up-complete means:Def39:
for X being non empty directed Subset of L
ex x being Element of L st x is_>=_than X &
for y being Element of L st y is_>=_than X holds x <= y;
end;
definition
cluster up-complete -> upper-bounded (with_suprema reflexive RelStr);
coherence
proof let L be with_suprema reflexive RelStr such that
A1: L is up-complete;
[#]L = the carrier of L by PRE_TOPC:12;
then ex x being Element of L st x is_>=_than the carrier of L &
for y being Element of L st y is_>=_than the carrier of L holds x <= y
by A1,Def39;
hence ex x being Element of L st x is_>=_than the carrier of L;
end;
end;
theorem
for L being non empty reflexive antisymmetric RelStr holds
L is up-complete
iff
for X being non empty directed Subset of L holds ex_sup_of X,L
proof let L be non empty reflexive antisymmetric RelStr;
hereby assume
A1: L is up-complete;
let X be non empty directed Subset of L;
consider x being Element of L such that
A2: x is_>=_than X and
A3: for y being Element of L st y is_>=_than X holds x <= y by A1,Def39;
thus ex_sup_of X,L
proof take x;
thus X is_<=_than x &
for b being Element of L st X is_<=_than b holds b >= x by A2,A3;
let c be Element of L; assume X is_<=_than c &
for b being Element of L st X is_<=_than b holds b >= c;
then c <= x & c >= x by A2,A3;
hence thesis by ORDERS_1:25;
end;
end;
assume
A4: for X being non empty directed Subset of L holds ex_sup_of X,L;
let X be non empty directed Subset of L;
ex_sup_of X,L by A4;
then ex a being Element of L st X is_<=_than a &
(for b being Element of L st X is_<=_than b holds b >= a) &
for c being Element of L st X is_<=_than c &
for b being Element of L st X is_<=_than b holds b >= c
holds c = a by YELLOW_0:def 7;
hence thesis;
end;
definition
let L be non empty reflexive RelStr;
attr L is /\-complete means:Def40:
for X being non empty Subset of L
ex x being Element of L st x is_<=_than X &
for y being Element of L st y is_<=_than X holds x >= y;
end;
theorem Th76:
for L being non empty reflexive antisymmetric RelStr holds
L is /\-complete
iff
for X being non empty Subset of L holds ex_inf_of X,L
proof let L be non empty reflexive antisymmetric RelStr;
hereby assume
A1: L is /\-complete;
let X be non empty Subset of L;
consider x being Element of L such that
A2: x is_<=_than X and
A3: for y being Element of L st y is_<=_than X holds x >= y by A1,Def40;
thus ex_inf_of X,L
proof take x;
thus X is_>=_than x &
for b being Element of L st X is_>=_than b holds b <= x by A2,A3;
let c be Element of L; assume X is_>=_than c &
for b being Element of L st X is_>=_than b holds b <= c;
then c <= x & c >= x by A2,A3;
hence thesis by ORDERS_1:25;
end;
end;
assume
A4: for X being non empty Subset of L holds ex_inf_of X,L;
let X be non empty Subset of L;
ex_inf_of X,L by A4;
then ex a being Element of L st X is_>=_than a &
(for b being Element of L st X is_>=_than b holds b <= a) &
for c being Element of L st X is_>=_than c &
for b being Element of L st X is_>=_than b holds b <= c
holds c = a by YELLOW_0:def 8;
hence thesis;
end;
definition
cluster complete -> up-complete /\-complete (non empty reflexive RelStr);
coherence
proof let L be non empty reflexive RelStr such that
A1: L is complete;
thus L is up-complete
proof let X be non empty directed Subset of L;
thus thesis by A1,LATTICE3:def 12;
end;
let X be non empty Subset of L;
set Y = {y where y is Element of L: y is_<=_than X};
consider x being Element of L such that
A2: Y is_<=_than x and
A3: for b being Element of L st Y is_<=_than b holds x <= b by A1,LATTICE3:def
12;
take x;
thus x is_<=_than X
proof let y be Element of L such that
A4: y in X;
y is_>=_than Y
proof let z be Element of L; assume z in Y;
then ex a being Element of L st z = a & a is_<=_than X;
hence thesis by A4,LATTICE3:def 8;
end;
hence x <= y by A3;
end;
let y be Element of L; assume
y is_<=_than X; then y in Y;
hence x >= y by A2,LATTICE3:def 9;
end;
cluster /\-complete -> lower-bounded (non empty reflexive RelStr);
coherence
proof let L be non empty reflexive RelStr; assume
L is /\-complete;
then [#]L = the carrier of L &
ex x being Element of L st x is_<=_than [#]L &
for y being Element of L st y is_<=_than [#]L holds x >= y
by Def40,PRE_TOPC:12;
hence ex x being Element of L st x is_<=_than the carrier of L;
end;
cluster up-complete with_suprema lower-bounded -> complete (non empty Poset);
coherence
proof let L be non empty Poset; assume
A5: L is up-complete with_suprema lower-bounded;
then reconsider K = L as with_suprema lower-bounded (non empty Poset);
let X be set;
X /\ the carrier of L c= the carrier of L by XBOOLE_1:17;
then reconsider X' = X /\ the carrier of L as Subset of L;
reconsider A = X' as Subset of K;
A6: now let Y be finite Subset of X';
Y c= the carrier of L by XBOOLE_1:1;
then Y is Subset of L;
hence Y <> {} implies ex_sup_of Y,L by A5,YELLOW_0:54;
end;
A7: finsups X' = {"\/"(Y,L) where Y is finite Subset of X': ex_sup_of Y,L}
by Def27;
A8: now let x be Element of L; assume x in finsups X';
then ex Y being finite Subset of X' st x = "\/"(Y,L) & ex_sup_of Y,L by
A7;
hence ex Y being finite Subset of X' st ex_sup_of Y,L & x = "\/"(Y,L);
end;
A9: now let Y be finite Subset of X';
Y c= the carrier of L by XBOOLE_1:1;
then reconsider Z = Y as Subset of L;
assume
Y <> {};
then ex_sup_of Z,L by A5,YELLOW_0:54;
hence "\/"(Y,L) in finsups X' by A7;
end;
reconsider fX = finsups A as non empty directed Subset of L;
consider x being Element of L such that
A10: fX is_<=_than x and
A11: for y being Element of L st fX is_<=_than y holds x <= y by A5,Def39;
take x;
X' is_<=_than x by A6,A8,A9,A10,Th52;
hence X is_<=_than x by YELLOW_0:5;
let y be Element of L;
assume y is_>=_than X;
then y is_>=_than X' by YELLOW_0:5;
then y is_>=_than fX by A6,A8,A9,Th52;
hence y >= x by A11;
end;
:: cluster up-complete /\-complete upper-bounded -> complete (non empty Poset);
:: to appear in YELLOW_2
end;
definition
cluster /\-complete -> with_infima (non empty reflexive antisymmetric RelStr);
coherence
proof let L be non empty reflexive antisymmetric RelStr;
assume L is /\-complete;
then for a,b being Element of L holds ex_inf_of {a,b},L by Th76;
hence L is with_infima by YELLOW_0:21;
end;
end;
definition
cluster /\-complete -> with_suprema
(non empty reflexive antisymmetric upper-bounded RelStr);
coherence
proof
let L be non empty reflexive antisymmetric upper-bounded RelStr such that
A1: L is /\-complete;
now let a,b be Element of L;
set X = {x where x is Element of L: x >= a & x >= b};
X c= the carrier of L
proof let x be set; assume x in X;
then ex z being Element of L st x = z & z >= a & z >= b;
hence thesis;
end;
then reconsider X as Subset of L;
Top L >= a & Top L >= b by YELLOW_0:45;
then Top L in X;
then ex_inf_of X,L by A1,Th76;
then consider c being Element of L such that
A2: c is_<=_than X and
A3: for d being Element of L st d is_<=_than X holds d <= c and
for e being Element of L st e is_<=_than X &
for d being Element of L st d is_<=_than X holds d <= e
holds e = c by YELLOW_0:def 8;
A4: c is_>=_than {a,b}
proof let x be Element of L; assume A5: x in {a,b};
x is_<=_than X
proof let y be Element of L; assume y in X;
then ex z being Element of L st y = z & z >= a & z >= b;
hence thesis by A5,TARSKI:def 2;
end;
hence thesis by A3;
end;
now let y be Element of L; assume
y is_>=_than {a,b};
then y >= a & y >= b by YELLOW_0:8;
then y in X;
hence c <= y by A2,LATTICE3:def 8;
end;
hence ex_sup_of {a,b},L by A4,YELLOW_0:15;
end;
hence thesis by YELLOW_0:20;
end;
end;
definition
cluster complete strict LATTICE;
existence
proof consider L being complete strict LATTICE;
take L; thus thesis;
end;
end;