Copyright (c) 1996 Association of Mizar Users
environ
vocabulary ORDERS_1, WAYBEL_0, LATTICE3, RELAT_2, YELLOW_0, BOOLE, LATTICES,
RELAT_1, WELLORD1, CAT_1, QUANTAL1, PRE_TOPC, FUNCT_1, GROUP_6, SEQM_3,
ORDINAL2, BINOP_1, BHSP_3, SETFAM_1, FILTER_2, YELLOW_1, SUBSET_1,
WELLORD2, YELLOW_2;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2,
SETFAM_1, LATTICE3, WELLORD1, STRUCT_0, ORDERS_1, PRE_TOPC, QUANTAL1,
ORDERS_3, YELLOW_0, YELLOW_1, WAYBEL_0, GRCAT_1;
constructors WAYBEL_0, YELLOW_1, TOLER_1, ORDERS_3, QUANTAL1, TOPS_2, GRCAT_1;
clusters STRUCT_0, LATTICE3, RELSET_1, YELLOW_0, YELLOW_1, WAYBEL_0, SUBSET_1,
XBOOLE_0;
requirements SUBSET, BOOLE;
definitions TARSKI, LATTICE3, QUANTAL1, ORDERS_3, WAYBEL_0;
theorems TARSKI, STRUCT_0, RELAT_1, FUNCT_1, FUNCT_2, SETFAM_1, LATTICE3,
WELLORD1, ORDERS_1, PRE_TOPC, TMAP_1, ORDERS_3, YELLOW_0, YELLOW_1,
WAYBEL_0, GRCAT_1, RELSET_1, XBOOLE_0, XBOOLE_1;
schemes FUNCT_2, SUPINF_2;
begin :: Basic Facts
reserve x, X, Y for set;
scheme RelStrSubset{L() -> non empty RelStr, P[set]}:
{x where x is Element of L(): P[x]} is Subset of L()
proof
{x where x is Element of L(): P[x]} c= the carrier of L()
proof
let x be set;
assume x in {l where l is Element of L(): P[l]};
then ex l being Element of L() st x = l & P[l];
hence thesis;
end;
hence thesis;
end;
theorem
for L being non empty RelStr
for x being Element of L
for X being Subset of L
holds
X c= downarrow x iff X is_<=_than x
proof let L be non empty RelStr, x be Element of L, X be Subset of L;
hereby
assume
A1: X c= downarrow x;
thus X is_<=_than x
proof let a be Element of L;
assume a in X;
hence a <= x by A1,WAYBEL_0:17;
end;
end;
assume
A2: for a being Element of L st a in X holds a <= x;
let a be set;
assume
A3: a in X;
then reconsider a as Element of L;
a <= x by A2,A3;
hence thesis by WAYBEL_0:17;
end;
theorem
for L being non empty RelStr
for x being Element of L
for X being Subset of L
holds
X c= uparrow x iff x is_<=_than X
proof let L be non empty RelStr, x be Element of L, X be Subset of L;
hereby
assume
A1: X c= uparrow x;
thus x is_<=_than X
proof let a be Element of L;
assume a in X;
hence x <= a by A1,WAYBEL_0:18;
end;
end;
assume
A2: for a being Element of L st a in X holds x <= a;
let a be set;
assume
A3: a in X;
then reconsider a as Element of L;
x <= a by A2,A3;
hence thesis by WAYBEL_0:18;
end;
theorem
for L being antisymmetric transitive with_suprema RelStr
for X, Y being set st ex_sup_of X,L & ex_sup_of Y,L
holds
ex_sup_of (X \/ Y),L & "\/"(X \/ Y, L) = "\/"(X, L) "\/" "\/"(Y, L)
proof
let L be antisymmetric transitive with_suprema RelStr;
let X, Y be set such that
A1: ex_sup_of X,L and
A2: ex_sup_of Y,L;
set a = "\/"(X, L) "\/" "\/"(Y, L);
A3: X \/ Y is_<=_than a
proof
let x be Element of L;
assume
A4: x in X \/ Y;
per cases by A4,XBOOLE_0:def 2;
suppose
A5: x in X;
X is_<=_than "\/"(X, L) by A1,YELLOW_0:30;
then x <= "\/"(X, L) & "\/"(X, L) <= a by A5,LATTICE3:def 9,YELLOW_0:22;
hence x <= a by ORDERS_1:26;
suppose
A6: x in Y;
Y is_<=_than "\/"(Y, L) by A2,YELLOW_0:30;
then x <= "\/"(Y, L) & "\/"(Y, L) <= a by A6,LATTICE3:def 9,YELLOW_0:22;
hence x <= a by ORDERS_1:26;
end;
for b being Element of L st X \/ Y is_<=_than b holds a <= b
proof
let b be Element of L;
assume
A7: X \/ Y is_<=_than b;
X c= X \/ Y & Y c= X \/ Y by XBOOLE_1:7;
then X is_<=_than b & Y is_<=_than b by A7,YELLOW_0:9;
then "\/"(X, L) <= b & "\/"(Y, L) <= b by A1,A2,YELLOW_0:30;
hence a <= b by YELLOW_0:22;
end;
hence thesis by A3,YELLOW_0:30;
end;
theorem
for L being antisymmetric transitive with_infima RelStr
for X, Y being set st ex_inf_of X,L & ex_inf_of Y,L
holds
ex_inf_of (X \/ Y),L & "/\"(X \/ Y, L) = "/\"(X, L) "/\" "/\"(Y, L)
proof
let L be antisymmetric transitive with_infima RelStr;
let X, Y be set such that
A1: ex_inf_of X,L and
A2: ex_inf_of Y,L;
set a = "/\"(X, L) "/\" "/\"(Y, L);
A3: X \/ Y is_>=_than a
proof
let x be Element of L;
assume
A4: x in X \/ Y;
per cases by A4,XBOOLE_0:def 2;
suppose
A5: x in X;
X is_>=_than "/\"(X, L) by A1,YELLOW_0:31;
then x >= "/\"(X, L) & "/\"(X, L) >= a by A5,LATTICE3:def 8,YELLOW_0:23;
hence x >= a by ORDERS_1:26;
suppose
A6: x in Y;
Y is_>=_than "/\"(Y, L) by A2,YELLOW_0:31;
then x >= "/\"(Y, L) & "/\"(Y, L) >= a by A6,LATTICE3:def 8,YELLOW_0:23;
hence x >= a by ORDERS_1:26;
end;
for b being Element of L st X \/ Y is_>=_than b holds a >= b
proof
let b be Element of L;
assume
A7: X \/ Y is_>=_than b;
X c= X \/ Y & Y c= X \/ Y by XBOOLE_1:7;
then X is_>=_than b & Y is_>=_than b by A7,YELLOW_0:9;
then "/\"(X, L) >= b & "/\"(Y, L) >= b by A1,A2,YELLOW_0:31;
hence a >= b by YELLOW_0:23;
end;
hence thesis by A3,YELLOW_0:31;
end;
begin :: Relational Substructures
theorem Th5:
for R being Relation
for X, Y being set
holds
X c= Y implies R |_2 X c= R |_2 Y
proof let R be Relation, X,Y be set;
assume
A1: X c= Y;
then X|R c= Y|R by RELAT_1:131;
then A2: X|R|X c= Y|R|X by RELAT_1:105;
Y|R|X c= Y|R|Y by A1,RELAT_1:104;
then X|R|X c= Y|R|Y by A2,XBOOLE_1:1;
then R|_2 X c= Y|R|Y by WELLORD1:17;
hence R|_2 X c= R|_2 Y by WELLORD1:17;
end;
theorem
for L being RelStr
for S, T being full SubRelStr of L st the carrier of S c= the carrier of T
holds
the InternalRel of S c= the InternalRel of T
proof
let L be RelStr, S1,S2 be full SubRelStr of L;
the InternalRel of S1 = (the InternalRel of L)|_2 the carrier of S1 &
the InternalRel of S2 = (the InternalRel of L)|_2 the carrier of S2
by YELLOW_0:def 14;
hence thesis by Th5;
end;
theorem Th7:
for L being non empty RelStr
for S being non empty SubRelStr of L
holds
(X is directed Subset of S implies X is directed Subset of L) &
(X is filtered Subset of S implies X is filtered Subset of L)
proof
let L be non empty RelStr;
let S be non empty SubRelStr of L;
thus X is directed Subset of S implies X is directed Subset of L
proof
assume
A1: X is directed Subset of S;
the carrier of S c= the carrier of L by YELLOW_0:def 13;
then X c= the carrier of L by A1,XBOOLE_1:1;
then reconsider X as Subset of L;
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
proof
let x, y be Element of L;
assume
A2: x in X & y in X;
then reconsider x'= x, y'= y as Element of S by A1;
consider z being Element of S such that
A3: z in X and
A4: x' <= z & y' <= z by A1,A2,WAYBEL_0:def 1;
reconsider z as Element of L by YELLOW_0:59;
take z;
thus thesis by A3,A4,YELLOW_0:60;
end;
hence thesis by WAYBEL_0:def 1;
end;
thus X is filtered Subset of S implies X is filtered Subset of L
proof
assume
A5: X is filtered Subset of S;
the carrier of S c= the carrier of L by YELLOW_0:def 13;
then X c= the carrier of L by A5,XBOOLE_1:1;
then reconsider X as Subset of L;
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
proof
let x, y be Element of L;
assume
A6: x in X & y in X;
then reconsider x'= x, y'= y as Element of S by A5;
consider z being Element of S such that
A7: z in X and
A8: z <= x' & z <= y' by A5,A6,WAYBEL_0:def 2;
reconsider z as Element of L by YELLOW_0:59;
take z;
thus thesis by A7,A8,YELLOW_0:60;
end;
hence thesis by WAYBEL_0:def 2;
end;
end;
theorem
for L being non empty RelStr
for S, T being non empty full SubRelStr of L
st the carrier of S c= the carrier of T
for X being Subset of S
holds
X is Subset of T & for Y being Subset of T st X = Y holds
(X is filtered implies Y is filtered) &
(X is directed implies Y is directed)
proof
let L be non empty RelStr, S1,S2 be non empty full SubRelStr of L;
assume
A1: the carrier of S1 c= the carrier of S2;
let X be Subset of S1;
X is Subset of S2 by A1,XBOOLE_1:1;
hence X is Subset of S2;
let X2 be Subset of S2 such that
A2: X = X2;
hereby assume
A3: X is filtered;
thus X2 is filtered
proof let x, y be Element of S2; assume
A4: x in X2 & y in X2;
then reconsider x'= x, y'= y as Element of S1 by A2;
consider z being Element of S1 such that
A5: z in X & z <= x' & z <= y' by A2,A3,A4,WAYBEL_0:def 2;
reconsider x1 = x, y1 = y, z1 = z as Element of L by YELLOW_0:59;
reconsider z as Element of S2 by A2,A5;
take z;
z1 <= x1 & z1 <= y1 by A5,YELLOW_0:60;
hence z in X2 & z <= x & z <= y by A2,A5,YELLOW_0:61;
end;
end;
assume
A6: X is directed;
let x, y be Element of S2; assume
A7: x in X2 & y in X2;
then reconsider x'= x, y'= y as Element of S1 by A2;
consider z being Element of S1 such that
A8: z in X & x' <= z & y' <= z by A2,A6,A7,WAYBEL_0:def 1;
reconsider x1 = x, y1 = y, z1 = z as Element of L by YELLOW_0:59;
reconsider z as Element of S2 by A2,A8;
take z;
x1 <= z1 & y1 <= z1 by A8,YELLOW_0:60;
hence z in X2 & x <= z & y <= z by A2,A8,YELLOW_0:61;
end;
begin :: Maps
scheme LambdaMD{X, Y() -> non empty RelStr, F(set) -> Element of Y()}:
ex f being map of X(), Y() st for x being Element of X() holds f.x = F(x)
proof deffunc f(set) = F($1);
consider f being Function of the carrier of X(),the carrier of Y() such that
A1: for s being Element of X() holds f.s = f(s)
from LambdaD;
reconsider f as map of X(),Y();
take f;
thus thesis by A1;
end;
scheme KappaMD{X, Y() -> non empty RelStr, F(set) -> set}:
ex f being map of X(), Y() st for x being Element of X() holds f.x = F(x)
provided
A1: for x being Element of X() holds F(x) is Element of Y()
proof
reconsider X = the carrier of X(),Y = the carrier of Y() as non empty set;
deffunc f(set) = F($1);
A2: now let x be Element of X;
F(x) is Element of Y() by A1;
hence f(x) in Y;
end;
consider f being Function of X, Y such that
A3: for x being Element of X holds f.x = f(x) from FunctR_ealEx(A2);
reconsider f as map of X(),Y();
take f;
thus thesis by A3;
end;
scheme NonUniqExMD{X, Y() -> non empty RelStr, P[set,set]}:
ex f being map of X(), Y() st for x being Element of X() holds P[x, f.x]
provided
A1: for x being Element of X() ex y being Element of Y() st P[x, y]
proof
defpred p[set,set] means P[$1,$2];
A2: for x being set st x in the carrier of X()
ex y being set st y in the carrier of Y() & p[x,y]
proof let x be set;
assume x in the carrier of X();
then reconsider x as Element of X();
ex y being Element of Y() st p[x,y] by A1;
hence thesis;
end;
consider f being Function of the carrier of X(),the carrier of Y() such that
A3: for x being set st x in the carrier of X() holds p[x,f.x]
from FuncEx1(A2);
reconsider f as map of X(),Y();
take f;
thus thesis by A3;
end;
definition
let S, T be 1-sorted;
let f be map of S, T;
redefine func rng f -> Subset of T;
coherence by RELSET_1:12;
end;
theorem Th9:
for S, T being non empty 1-sorted
for f, g being map of S, T
holds
(for s being Element of S holds f.s = g.s) implies f = g by FUNCT_2:113;
definition
let J be set, L be RelStr;
let f, g be Function of J, the carrier of L;
pred f <= g means
:Def1:
for j being set st j in J
ex a, b being Element of L st a = f.j & b = g.j & a <= b;
synonym g >= f;
end;
theorem
for L, M being non empty RelStr, f,g being map of L, M holds
f <= g iff for x being Element of L holds f.x <= g.x
proof let L, M be non empty RelStr, f,g be map of L, M;
hereby assume
A1: f <= g;
let x be Element of L;
ex m1,m2 being Element of M st m1 = f.x & m2 = g.x & m1 <= m2
by A1,Def1;
hence f.x <= g.x;
end;
assume
A2: for x being Element of L holds f.x <= g.x;
let x be set;
assume x in the carrier of L;
then reconsider x as Element of L;
take f.x, g.x;
thus thesis by A2;
end;
begin :: The Image of a Map
definition
let L, M be non empty RelStr;
let f be map of L, M;
func Image f -> strict full SubRelStr of M equals
:Def2:
subrelstr rng f;
correctness;
end;
theorem Th11:
for L, M being non empty RelStr
for f being map of L, M
holds
rng f = the carrier of Image f
proof let L1, L2 be non empty RelStr, g be map of L1,L2;
Image g = subrelstr(rng g) by Def2;
hence the carrier of Image g = rng g by YELLOW_0:def 15;
end;
theorem
for L, M being non empty RelStr
for f being map of L, M
for y being Element of Image f
ex x being Element of L st f.x = y
proof let L1, L2 be non empty RelStr, g be map of L1,L2;
let s be Element of Image g;
A1: dom g = the carrier of L1 by FUNCT_2:def 1;
then rng g is non empty & rng g = the carrier of Image g by Th11,RELAT_1:65;
then consider l being set such that
A2: l in dom g and
A3: s = g.l by FUNCT_1:def 5;
reconsider l as Element of L1 by A1,A2;
take l;
thus thesis by A3;
end;
definition
let L be non empty RelStr;
let X be non empty Subset of L;
cluster subrelstr X -> non empty;
coherence
proof the carrier of subrelstr X = X by YELLOW_0:def 15;
hence thesis by STRUCT_0:def 1;
end;
end;
definition
let L, M be non empty RelStr;
let f be map of L, M;
cluster Image f -> non empty;
coherence
proof
dom f = the carrier of L by FUNCT_2:def 1;
then rng f is non empty & rng f = the carrier of Image f by Th11,RELAT_1:65;
hence Image f is non empty by STRUCT_0:def 1;
end;
end;
begin :: Monotone Maps
theorem
for L being non empty RelStr holds id L is monotone
proof
let L be non empty RelStr;
let l1,l2 be Element of L;
assume l1 <= l2;
then l1 <= (id L).l2 by TMAP_1:91;
hence thesis by TMAP_1:91;
end;
theorem
for L, M, N being non empty RelStr
for f being map of L, M, g being map of M, N
holds
f is monotone & g is monotone implies g*f is monotone
proof let L1,L2,L3 be non empty RelStr;
let g1 be map of L1,L2, g2 be map of L2,L3 such that
A1: g1 is monotone and
A2: g2 is monotone;
let s1,s2 be Element of L1;
assume s1 <= s2;
then g1.s1 <= g1.s2 by A1,ORDERS_3:def 5;
then g2.(g1.s1) <= g2.(g1.s2) by A2,ORDERS_3:def 5;
then (g2*g1).s1 <= g2.(g1.s2) by FUNCT_2:21;
hence thesis by FUNCT_2:21;
end;
theorem
for L, M being non empty RelStr
for f being map of L, M
for X being Subset of L, x being Element of L
holds
f is monotone & x is_<=_than X implies f.x is_<=_than f.:X
proof let L1,L2 be non empty RelStr, g be map of L1,L2;
let X be Subset of L1, x be Element of L1 such that
A1: g is monotone and
A2: x is_<=_than X;
let y2 be Element of L2; assume
y2 in g.:X;
then consider x2 being Element of L1 such that
A3: x2 in X and
A4: y2 = g.x2 by FUNCT_2:116;
reconsider x2 as Element of L1;
x <= x2 by A2,A3,LATTICE3:def 8;
hence g.x <= y2 by A1,A4,ORDERS_3:def 5;
end;
theorem
for L, M being non empty RelStr
for f being map of L, M
for X being Subset of L, x being Element of L
holds
f is monotone & X is_<=_than x implies f.:X is_<=_than f.x
proof let L1,L2 be non empty RelStr, g be map of L1,L2;
let X be Subset of L1, x be Element of L1 such that
A1: g is monotone and
A2: x is_>=_than X;
let y2 be Element of L2; assume
y2 in g.:X;
then consider x2 being Element of L1 such that
A3: x2 in X and
A4: y2 = g.x2 by FUNCT_2:116;
reconsider x2 as Element of L1;
x >= x2 by A2,A3,LATTICE3:def 9;
hence g.x >= y2 by A1,A4,ORDERS_3:def 5;
end;
theorem
for S, T being non empty RelStr
for f being map of S, T
for X being directed Subset of S
holds
f is monotone implies f.:X is directed
proof
let S, T be non empty RelStr;
let f be map of S, T;
let X be directed Subset of S;
assume
A1: f is monotone;
set Y = f.:X;
for y1, y2 being Element of T st y1 in Y & y2 in Y
ex z being Element of T st z in Y & y1 <= z & y2 <= z
proof
let y1, y2 be Element of T;
assume
A2: y1 in Y & y2 in Y;
then consider x1 being set such that
x1 in dom f and
A3: x1 in X and
A4: y1 = f.x1 by FUNCT_1:def 12;
consider x2 being set such that
x2 in dom f and
A5: x2 in X and
A6: y2 = f.x2 by A2,FUNCT_1:def 12;
reconsider x1, x2 as Element of S by A3,A5;
consider z being Element of S such that
A7: z in X & x1 <= z & x2 <= z by A3,A5,WAYBEL_0:def 1;
take f.z;
thus f.z in Y by A7,FUNCT_2:43;
thus y1 <= f.z & y2 <= f.z by A1,A4,A6,A7,ORDERS_3:def 5;
end;
hence f.:X is directed by WAYBEL_0:def 1;
end;
theorem Th18:
for L being with_suprema Poset
for f being map of L, L
holds
f is directed-sups-preserving implies f is monotone
proof
let L be with_suprema Poset;
let f be map of L, L;
assume
A1: f is directed-sups-preserving;
let x, y be Element of L such that
A2: x <= y;
let afx, bfy be Element of L such that
A3: afx = f.x & bfy = f.y;
A4: y = y"\/"x by A2,YELLOW_0:24;
x <= y & y <= y by A2;
then A5: {x, y} is_<=_than y by YELLOW_0:8;
for b being Element of L st {x, y} is_<=_than b holds y <= b by YELLOW_0:8;
then A6: ex_sup_of {x, y},L by A5,YELLOW_0:30;
for a, b being Element of L st a in {x, y} & b in {x, y}
ex z being Element of L st z in {x, y} & a <= z & b <= z
proof
let a, b be Element of L such that
A7: a in {x, y} & b in {x, y};
take y;
thus y in {x, y} by TARSKI:def 2;
thus a <= y & b <= y by A2,A7,TARSKI:def 2;
end;
then {x, y} is directed non empty by WAYBEL_0:def 1;
then f preserves_sup_of {x, y} by A1,WAYBEL_0:def 37;
then A8: sup(f.:{x, y}) = f.sup{x, y} by A6,WAYBEL_0:def 31
.= f.y by A4,YELLOW_0:41;
dom f = the carrier of L by FUNCT_2:def 1;
then f.y = sup{f.x, f.y} by A8,FUNCT_1:118
.= f.y"\/"f.x by YELLOW_0:41;
hence afx <= bfy by A3,YELLOW_0:22;
end;
theorem
for L being with_infima Poset
for f being map of L, L
holds
f is filtered-infs-preserving implies f is monotone
proof
let L be with_infima Poset;
let f be map of L, L;
assume
A1: f is filtered-infs-preserving;
let x, y be Element of L such that
A2: x <= y;
let a, b be Element of L such that
A3: a = f.x & b = f.y;
A4: x = x"/\"y by A2,YELLOW_0:25;
x <= x & x <= y by A2;
then A5: x is_<=_than {x, y} by YELLOW_0:8;
for c being Element of L st c is_<=_than {x, y} holds c <= x by YELLOW_0:8
;
then A6: ex_inf_of {x, y},L by A5,YELLOW_0:31;
for c, d being Element of L st c in {x, y} & d in {x, y}
ex z being Element of L st z in {x, y} & z <= c & z <= d
proof
let c, d be Element of L such that
A7: c in {x, y} & d in {x, y};
take x;
thus x in {x, y} by TARSKI:def 2;
thus x <= c & x <= d by A2,A7,TARSKI:def 2;
end;
then {x, y} is filtered non empty by WAYBEL_0:def 2;
then f preserves_inf_of {x, y} by A1,WAYBEL_0:def 36;
then A8: inf(f.:{x, y}) = f.inf{x, y} by A6,WAYBEL_0:def 30
.= f.x by A4,YELLOW_0:40;
dom f = the carrier of L by FUNCT_2:def 1;
then f.x = inf{f.x, f.y} by A8,FUNCT_1:118
.= f.x"/\"f.y by YELLOW_0:40;
hence a <= b by A3,YELLOW_0:23;
end;
begin :: Idempotent Maps
theorem
for S being non empty 1-sorted
for f being map of S, S
holds
f is idempotent implies for x being Element of S holds f.(f.x) = f.x
proof let L be non empty 1-sorted, p be map of L,L; assume
A1: p*p = p;
let l be Element of L;
thus p.(p.l) = p.l by A1,FUNCT_2:21;
end;
theorem Th21:
for S being non empty 1-sorted
for f being map of S, S
holds
f is idempotent implies rng f = {x where x is Element of S: x = f.x}
proof
let S be non empty 1-sorted;
let f be map of S, S;
assume
A1: f = f*f;
set M = {x where x is Element of S: x = f.x};
A2:
rng f c= M
proof
let y be set;
assume
A3: y in rng f;
then consider x being set such that
A4: x in dom f and
A5: y = f.x by FUNCT_1:def 5;
reconsider y'= y as Element of S by A3;
y'= f.y' by A1,A4,A5,FUNCT_1:23;
hence y in M;
end;
M c= rng f
proof
let y be set;
assume y in M;
then consider y' being Element of S such that
A6: y' = y and
A7: y' = f.y';
dom f = the carrier of S by FUNCT_2:def 1;
hence y in rng f by A6,A7,FUNCT_1:def 5;
end;
hence rng f = {x where x is Element of S: x = f.x} by A2,XBOOLE_0:def 10;
end;
theorem Th22:
for S being non empty 1-sorted
for f being map of S, S st f is idempotent
holds
X c= rng f implies f.:X = X
proof
let S be non empty 1-sorted;
let f be map of S, S such that
A1: f is idempotent;
set M = {x where x is Element of S: x = f.x};
assume X c= rng f;
then A2: X c= M by A1,Th21;
A3:
f.:X c= X
proof
let y be set;
assume y in f.:X;
then consider x being set such that
x in dom f and
A4: x in X and
A5: y = f.x by FUNCT_1:def 12;
x in M by A2,A4;
then consider x' being Element of S such that
A6: x' = x and
A7: x' = f.x';
thus y in X by A4,A5,A6,A7;
end;
X c= f.:X
proof
let y be set;
assume
A8: y in X;
then y in M by A2;
then consider x being Element of S such that
A9: x = y and
A10: x = f.x;
thus y in f.:X by A8,A9,A10,FUNCT_2:43;
end;
hence f.:X = X by A3,XBOOLE_0:def 10;
end;
theorem
for L being non empty RelStr holds id L is idempotent
proof
let L be non empty RelStr;
id L = id the carrier of L by GRCAT_1:def 11;
hence (id L)*(id L)
= id ((the carrier of L) /\ (the carrier of L)) by FUNCT_1:43
.= (id L) by GRCAT_1:def 11;
end;
begin :: Complete Lattices (CCL Chapter 0, Section 2, pp. 8 -- 9)
reserve L for complete LATTICE,
a for Element of L;
theorem Th24:
a in X implies a <= "\/"(X, L) & "/\"(X, L) <= a
proof
assume
A1: a in X;
X is_<=_than "\/"(X, L) by YELLOW_0:32;
hence a <= "\/"(X, L) by A1,LATTICE3:def 9;
X is_>=_than "/\"(X, L) by YELLOW_0:33;
hence "/\"(X, L) <= a by A1,LATTICE3:def 8;
end;
theorem Th25:
for L being (non empty RelStr)
holds
(for X holds ex_sup_of X,L) iff (for Y holds ex_inf_of Y,L)
proof
let L be (non empty RelStr);
hereby
assume
A1: for X holds ex_sup_of X,L;
let Y;
set X = {x where x is Element of L: x is_<=_than Y};
ex_sup_of X,L by A1;
then consider a being Element of L such that
A2: X is_<=_than a and
A3: for b being Element of L st X is_<=_than b holds b >= a and
A4: 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;
A5: a is_<=_than Y
proof
let b be Element of L;
assume
A6: b in Y;
b is_>=_than X
proof
let c be Element of L;
assume c in X;
then ex x being Element of L st c = x & x is_<=_than Y;
hence c <= b by A6,LATTICE3:def 8;
end;
hence a <= b by A3;
end;
A7: for b being Element of L st Y is_>=_than b holds a >= b
proof
let b be Element of L;
assume b is_<=_than Y;
then b in X;
hence b <= a by A2,LATTICE3:def 9;
end;
for c being Element of L st Y is_>=_than c &
for b being Element of L st Y is_>=_than b holds b <= c
holds c = a
proof
let c be Element of L such that
A8: Y is_>=_than c and
A9: for b being Element of L st Y is_>=_than b holds b <= c;
A10: X is_<=_than c
proof
let b be Element of L;
assume b in X;
then ex x being Element of L st b = x & x is_<=_than Y;
hence b <= c by A9;
end;
for b being Element of L st X is_<=_than b holds b >= c
proof
let b be Element of L;
assume
A11: X is_<=_than b;
c in X by A8;
hence c <= b by A11,LATTICE3:def 9;
end;
hence thesis by A4,A10;
end;
hence ex_inf_of Y,L by A5,A7,YELLOW_0:def 8;
end;
assume
A12: for Y holds ex_inf_of Y,L;
let X;
set Y = {y where y is Element of L: X is_<=_than y};
ex_inf_of Y,L by A12;
then consider a being Element of L such that
A13: Y is_>=_than a and
A14: for b being Element of L st Y is_>=_than b holds b <= a and
A15: for c being Element of L st Y is_>=_than c &
for b being Element of L st Y is_>=_than b holds b <= c
holds c = a by YELLOW_0:def 8;
A16: X is_<=_than a
proof
let b be Element of L;
assume
A17: b in X;
b is_<=_than Y
proof
let c be Element of L;
assume c in Y;
then ex y being Element of L st c = y & X is_<=_than y;
hence b <= c by A17,LATTICE3:def 9;
end;
hence b <= a by A14;
end;
A18: for b being Element of L st X is_<=_than b holds a <= b
proof
let b be Element of L;
assume X is_<=_than b;
then b in Y;
hence a <= b by A13,LATTICE3:def 8;
end;
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
proof
let c be Element of L such that
A19: X is_<=_than c and
A20: for b being Element of L st X is_<=_than b holds b >= c;
A21: Y is_>=_than c
proof
let b be Element of L;
assume b in Y;
then ex x being Element of L st b = x & x is_>=_than X;
hence c <= b by A20;
end;
for b being Element of L st Y is_>=_than b holds b <= c
proof
let b be Element of L;
assume
A22: Y is_>=_than b;
c in Y by A19;
hence b <= c by A22,LATTICE3:def 8;
end;
hence c = a by A15,A21;
end;
hence ex_sup_of X,L by A16,A18,YELLOW_0:def 7;
end;
theorem Th26: ::Proposition 2.2 (i) (variant 1) cf YELLOW_0
for L being (non empty RelStr)
holds
(for X holds ex_sup_of X,L) implies L is complete
proof
let L be (non empty RelStr);
assume
A1: for X holds ex_sup_of X,L;
L is complete
proof
let X be set;
take a = "\/"(X, L);
A2: ex_sup_of X,L by A1;
hence X is_<=_than a by YELLOW_0:def 9;
thus thesis by A2,YELLOW_0:def 9;
end;
hence thesis;
end;
theorem Th27: ::Proposition 2.2 (i) (variant 2) cf variant 3
for L being (non empty RelStr)
holds
(for X holds ex_inf_of X,L) implies L is complete
proof
let L be (non empty RelStr);
assume for X holds ex_inf_of X,L;
then for X holds ex_sup_of X,L by Th25;
hence thesis by Th26;
end;
theorem Th28:
for L being (non empty RelStr)
holds
(for A being Subset of L holds ex_inf_of A, L) implies
for X holds ex_inf_of X,L & "/\"(X, L) = "/\"(X /\ the carrier of L, L)
proof
let L be (non empty RelStr);
assume
A1: for A being Subset of L holds ex_inf_of A, L;
let X;
set Y = X /\ the carrier of L;
set a = "/\"(Y, L);
Y c= the carrier of L by XBOOLE_1:17;
then reconsider Y as Subset of L;
A2: ex_inf_of Y,L by A1;
then A3: ex_inf_of X,L by YELLOW_0:50;
A4:
a is_<=_than X
proof
let x be Element of L;
assume x in X;
then A5: x in Y by XBOOLE_0:def 3;
a is_<=_than Y by A2,YELLOW_0:def 10;
hence a <= x by A5,LATTICE3:def 8;
end;
for b being Element of L st b is_<=_than X holds b <= a
proof
let b be Element of L;
assume
A6: b is_<=_than X;
Y c= X by XBOOLE_1:17;
then b is_<=_than Y by A6,YELLOW_0:9;
hence b <= a by A2,YELLOW_0:def 10;
end;
hence thesis by A3,A4,YELLOW_0:def 10;
end;
theorem
for L being (non empty RelStr)
holds
(for A being Subset of L holds ex_sup_of A, L) implies
for X holds ex_sup_of X,L & "\/"(X, L) = "\/"(X /\ the carrier of L, L)
proof
let L be (non empty RelStr);
assume
A1: for A being Subset of L holds ex_sup_of A, L;
let X;
set Y = X /\ the carrier of L;
set a = "\/"(Y, L);
Y c= the carrier of L by XBOOLE_1:17;
then reconsider Y as Subset of L;
A2: ex_sup_of Y,L by A1;
then A3: ex_sup_of X,L by YELLOW_0:50;
A4:
X is_<=_than a
proof
let x be Element of L;
assume x in X;
then A5: x in Y by XBOOLE_0:def 3;
Y is_<=_than a by A2,YELLOW_0:def 9;
hence x <= a by A5,LATTICE3:def 9;
end;
for b being Element of L st X is_<=_than b holds a <= b
proof
let b be Element of L;
assume
A6: X is_<=_than b;
Y c= X by XBOOLE_1:17;
then Y is_<=_than b by A6,YELLOW_0:9;
hence a <= b by A2,YELLOW_0:def 9;
end;
hence thesis by A3,A4,YELLOW_0:def 9;
end;
theorem Th30: ::Proposition 2.2 (i) (variant 3)
for L being (non empty RelStr)
holds
(for A being Subset of L holds ex_inf_of A,L) implies L is complete
proof
let L be (non empty RelStr);
assume for A being Subset of L holds ex_inf_of A,L;
then for X holds ex_inf_of X, L by Th28;
hence thesis by Th27;
end;
Lm1:
for L being non empty Poset
holds
L is up-complete /\-complete upper-bounded implies
L is complete (non empty Poset)
proof
let L be non empty Poset;
assume
A1: L is up-complete /\-complete upper-bounded;
for A being Subset of L holds ex_inf_of A,L
proof
let A be Subset of L;
per cases;
suppose A is empty;
hence ex_inf_of A,L by A1,YELLOW_0:43;
suppose A is non empty;
hence ex_inf_of A,L by A1,WAYBEL_0:76;
end;
hence thesis by Th30;
end;
definition
cluster up-complete /\-complete upper-bounded -> complete (non empty Poset);
correctness by Lm1;
end;
theorem Th31: :: Theorem 2.3 (The Fixed-Point Theorem)
for f being map of L, L st f is monotone
for M being Subset of L st M = {x where x is Element of L: x = f.x}
holds
subrelstr M is complete LATTICE
proof
let f be map of L, L such that
A1: f is monotone;
let M be Subset of L such that
A2: M = {x where x is Element of L: x = f.x};
set S = subrelstr M;
A3:
for X being Subset of S
for Y being Subset of L st
Y = {y where y is Element of L: X is_<=_than f.y & f.y <= y}
holds
inf Y in M
proof
let X be Subset of S;
let Y be Subset of L such that
A4: Y = {y where y is Element of L: X is_<=_than f.y & f.y <= y};
set z = inf Y;
A5: ex_inf_of Y,L by YELLOW_0:17;
f.z is_<=_than Y
proof
let u be Element of L;
assume
A6: u in Y;
then consider y being Element of L such that
A7: y = u and
X is_<=_than f.y and
A8: f.y <= y by A4;
z <= u by A6,Th24;
then f.z <= f.y by A1,A7,ORDERS_3:def 5;
hence f.z <= u by A7,A8,ORDERS_1:26;
end;
then A9: f.z <= z by A5,YELLOW_0:31;
then A10: f.(f.z) <= f.z by A1,ORDERS_3:def 5;
X is_<=_than f.(f.z)
proof
let m be Element of L;
assume
A11: m in X;
X c= the carrier of S;
then X c= M by YELLOW_0:def 15;
then m in M by A11;
then consider x being Element of L such that
A12: x = m and
A13: x = f.x by A2;
m is_<=_than Y
proof
let u be Element of L;
assume u in Y;
then consider y being Element of L such that
A14: y = u and
A15: X is_<=_than f.y and
A16: f.y <= y by A4;
m <= f.y by A11,A15,LATTICE3:def 9;
hence m <= u by A14,A16,ORDERS_1:26;
end;
then m <= z by YELLOW_0:33;
then f.m <= f.z by A1,ORDERS_3:def 5;
hence m <= f.(f.z) by A1,A12,A13,ORDERS_3:def 5;
end;
then f.z in Y by A4,A10;
then z <= f.z by Th24;
then z = f.z by A9,ORDERS_1:25;
hence inf Y in M by A2;
end;
M c= the carrier of S by YELLOW_0:def 15;
then reconsider M as Subset of S;
defpred P[Element of L] means M is_<=_than f.$1 & f.$1 <= $1;
reconsider Y = {y where y is Element of L: P[y]} as Subset of L
from RelStrSubset;
inf Y in M by A3;
then reconsider S as non empty Poset by STRUCT_0:def 1;
for X being Subset of S holds ex_sup_of X, S
proof
let X be Subset of S;
defpred Q[Element of L] means X is_<=_than f.$1 & f.$1 <= $1;
reconsider Y = {y where y is Element of L: Q[y]} as Subset of L
from RelStrSubset;
set z = inf Y;
z in M & M = the carrier of S by A3,YELLOW_0:def 15;
then reconsider z as Element of S;
A17:
X is_<=_than z
proof
let x be Element of S;
assume
A18: x in X;
x in the carrier of S;
then x in M by YELLOW_0:def 15;
then consider m being Element of L such that
A19: x = m and
m = f.m by A2;
m is_<=_than Y
proof
let u be Element of L;
assume u in Y;
then consider y being Element of L such that
A20: y = u and
A21: X is_<=_than f.y and
A22: f.y <= y;
m <= f.y by A18,A19,A21,LATTICE3:def 9;
hence m <= u by A20,A22,ORDERS_1:26;
end;
then m <= inf Y & x in the carrier of S & z in
the carrier of S by YELLOW_0:33;
hence x <= z by A19,YELLOW_0:61;
end;
for b being Element of S st X is_<=_than b holds z <= b
proof
let b be Element of S;
b in the carrier of S;
then b in M by YELLOW_0:def 15;
then consider x being Element of L such that
A23: x = b and
A24: x = f.x by A2;
assume
X is_<=_than b;
then X is_<=_than f.x & f.x <= x by A23,A24,YELLOW_0:63;
then x in Y;
then inf Y <= x & z in the carrier of S & b in the carrier of S by Th24;
hence z <= b by A23,YELLOW_0:61;
end;
hence thesis by A17,YELLOW_0:30;
end;
then reconsider S as complete (non empty Poset) by YELLOW_0:53;
S is complete LATTICE;
hence thesis;
end;
theorem Th32:
for S being infs-inheriting non empty full SubRelStr of L
holds
S is complete LATTICE
proof
let S be infs-inheriting non empty full SubRelStr of L;
for X being Subset of S holds ex_inf_of X,S
proof
let X be Subset of S;
A1: ex_inf_of X,L by YELLOW_0:17;
then "/\"(X,L) in the carrier of S by YELLOW_0:def 18;
hence ex_inf_of X,S by A1,YELLOW_0:64;
end;
then S is complete by Th30;
hence S is complete LATTICE by LATTICE3:12;
end;
theorem Th33:
for S being sups-inheriting non empty full SubRelStr of L
holds
S is complete LATTICE
proof
let S be sups-inheriting non empty full SubRelStr of L;
for X being Subset of S holds ex_sup_of X, S
proof
let X be Subset of S;
A1: ex_sup_of X,L by YELLOW_0:17;
then "\/"(X,L) in the carrier of S by YELLOW_0:def 19;
hence ex_sup_of X,S by A1,YELLOW_0:65;
end;
then S is complete by YELLOW_0:53;
hence S is complete LATTICE by LATTICE3:12;
end;
theorem Th34: ::Remark 2.4 (Part I, variant 1)
for M being non empty RelStr
for f being map of L, M st f is sups-preserving
holds
Image f is sups-inheriting
proof
let M be non empty RelStr;
let f be map of L, M such that
A1: f is sups-preserving;
set S = subrelstr(rng f);
for Y being Subset of S st ex_sup_of Y,M holds "\/"(Y, M) in the carrier of
S
proof
let Y be Subset of S;
assume ex_sup_of Y,M;
A2: f preserves_sup_of (f"Y) by A1,WAYBEL_0:def 33;
A3: ex_sup_of f"Y,L by YELLOW_0:17;
Y c= the carrier of S;
then Y c= rng f by YELLOW_0:def 15;
then "\/"(Y, M) = sup(f.:(f"Y)) by FUNCT_1:147
.= f.sup(f"Y) by A2,A3,WAYBEL_0:def 31;
then "\/"(Y, M) in rng f by FUNCT_2:6;
hence "\/"(Y, M) in the carrier of S by YELLOW_0:def 15;
end;
then S is sups-inheriting by YELLOW_0:def 19;
hence thesis by Def2;
end;
theorem Th35: ::Remark 2.4 (Part I, variant 2)
for M being non empty RelStr
for f being map of L, M st f is infs-preserving
holds
Image f is infs-inheriting
proof
let M be non empty RelStr;
let f be map of L, M such that
A1: f is infs-preserving;
set S = subrelstr(rng f);
for Y being Subset of S st ex_inf_of Y,M holds "/\"(Y, M) in the carrier of
S
proof
let Y be Subset of S;
assume ex_inf_of Y,M;
A2: f preserves_inf_of (f"Y) by A1,WAYBEL_0:def 32;
A3: ex_inf_of f"Y,L by YELLOW_0:17;
Y c= the carrier of S;
then Y c= rng f by YELLOW_0:def 15;
then "/\"(Y, M) = inf(f.:(f"Y)) by FUNCT_1:147
.= f.inf(f"Y) by A2,A3,WAYBEL_0:def 30;
then "/\"(Y, M) in rng f by FUNCT_2:6;
hence "/\"(Y, M) in the carrier of S by YELLOW_0:def 15;
end;
then S is infs-inheriting by YELLOW_0:def 18;
hence thesis by Def2;
end;
theorem ::Remark 2.4 (Part II)
for L, M being complete LATTICE
for f being map of L, M st f is sups-preserving or f is infs-preserving
holds
Image f is complete LATTICE
proof
let L, M be complete LATTICE;
let f be map of L, M such that
A1: f is sups-preserving or f is infs-preserving;
per cases by A1;
suppose f is sups-preserving;
then Image f is sups-inheriting by Th34;
hence Image f is complete LATTICE by Th33;
suppose f is infs-preserving;
then Image f is infs-inheriting by Th35;
hence Image f is complete LATTICE by Th32;
end;
theorem ::Remark 2.5
for f being map of L, L st f is idempotent directed-sups-preserving
holds
Image f is directed-sups-inheriting & Image f is complete LATTICE
proof
let f be map of L, L;
assume
A1: f is idempotent directed-sups-preserving;
then A2: f is monotone by Th18;
set S = subrelstr(rng f);
consider a being Element of L;
dom f = the carrier of L by FUNCT_2:def 1;
then f.a in rng f by FUNCT_1:def 5;
then the carrier of S is non empty by YELLOW_0:def 15;
then reconsider S'= S as non empty full SubRelStr of L by STRUCT_0:def 1;
for X being directed Subset of S st X <> {} & ex_sup_of X,L
holds "\/"(X,L) in the carrier of S
proof
let X be directed Subset of S;
assume X <> {};
then X is non empty directed Subset of S';
then reconsider X'= X as non empty directed Subset of L by Th7;
A3: f preserves_sup_of X' by A1,WAYBEL_0:def 37;
assume ex_sup_of X,L;
then A4: ex_sup_of f.:X',L & sup(f.:X') = f.sup X' by A3,WAYBEL_0:def 31;
X c= the carrier of S;
then X c= rng f by YELLOW_0:def 15;
then sup X' = f.sup X' & the carrier of L <> {} by A1,A4,Th22;
then "\/"(X, L) in rng f by FUNCT_2:6;
hence "\/"(X, L) in the carrier of S by YELLOW_0:def 15;
end;
then S is directed-sups-inheriting by WAYBEL_0:def 4;
hence Image f is directed-sups-inheriting by Def2;
rng f = {x where x is Element of L: x = f.x} by A1,Th21;
then S is complete LATTICE by A2,Th31;
hence Image f is complete LATTICE by Def2;
end;
begin :: A Lattice of Ideals
theorem Th38:
for L being RelStr
for F being Subset of bool the carrier of L st
for X being Subset of L st X in F holds X is lower
holds meet F is lower Subset of L
proof
let L be RelStr;
let F be Subset of bool the carrier of L;
reconsider F' = F as Subset-Family of L by SETFAM_1:def 7;
assume
A1: for X being Subset of L st X in F holds X is lower;
reconsider M = meet F' as Subset of L;
per cases;
suppose
F = {};
then for x, y being Element of L st x in M & y <= x holds y in M
by SETFAM_1:def 1;
hence thesis by WAYBEL_0:def 19;
suppose
A2: F <> {};
for x, y being Element of L st x in M & y <= x holds y in M
proof
let x, y be Element of L;
assume that
A3: x in M and
A4: y <= x;
for Y being set st Y in F holds y in Y
proof
let Y be set;
assume
A5: Y in F;
then reconsider X = Y as Subset of L;
A6: X is lower by A1,A5;
x in X by A3,A5,SETFAM_1:def 1;
hence y in Y by A4,A6,WAYBEL_0:def 19;
end;
hence y in M by A2,SETFAM_1:def 1;
end;
hence thesis by WAYBEL_0:def 19;
end;
theorem
for L being RelStr
for F being Subset of bool the carrier of L st
for X being Subset of L st X in F holds X is upper
holds meet F is upper Subset of L
proof
let L be RelStr;
let F be Subset of bool the carrier of L;
reconsider F' = F as Subset-Family of L by SETFAM_1:def 7;
assume
A1: for X being Subset of L st X in F holds X is upper;
reconsider M = meet F' as Subset of L;
per cases;
suppose
F = {};
then for x, y being Element of L st x in M & x <= y holds y in M
by SETFAM_1:def 1;
hence thesis by WAYBEL_0:def 20;
suppose
A2: F <> {};
for x, y being Element of L st x in M & x <= y holds y in M
proof
let x, y be Element of L;
assume that
A3: x in M and
A4: x <= y;
for Y being set st Y in F holds y in Y
proof
let Y be set;
assume
A5: Y in F;
then reconsider X = Y as Subset of L;
A6: X is upper by A1,A5;
x in X by A3,A5,SETFAM_1:def 1;
hence y in Y by A4,A6,WAYBEL_0:def 20;
end;
hence y in M by A2,SETFAM_1:def 1;
end;
hence thesis by WAYBEL_0:def 20;
end;
theorem Th40:
for L being with_suprema antisymmetric RelStr
for F being Subset of bool the carrier of L st
for X being Subset of L st X in F holds X is lower directed
holds meet F is directed Subset of L
proof
let L be with_suprema antisymmetric RelStr;
let F be Subset of bool the carrier of L;
assume
A1: for X being Subset of L st X in F holds X is lower directed;
reconsider F' = F as Subset-Family of L by SETFAM_1:def 7;
reconsider M = meet F' as Subset of L;
per cases;
suppose
A2: F = {};
M is directed
proof
let x, y be Element of L;
assume x in M & y in M;
hence thesis by A2,SETFAM_1:def 1;
end;
hence thesis;
suppose
A3: F <> {};
M is directed
proof
let x, y be Element of L such that
A4: x in M & y in M;
take z = x"\/"y;
for Y being set st Y in F holds z in Y
proof
let Y be set;
assume
A5: Y in F;
then reconsider X = Y as Subset of L;
A6: X is lower directed by A1,A5;
x in X & y in X by A4,A5,SETFAM_1:def 1;
hence z in Y by A6,WAYBEL_0:40;
end;
hence z in M by A3,SETFAM_1:def 1;
thus x <= z & y <= z by YELLOW_0:22;
end;
hence thesis;
end;
theorem
for L being with_infima antisymmetric RelStr
for F being Subset of bool the carrier of L st
for X being Subset of L st X in F holds X is upper filtered
holds meet F is filtered Subset of L
proof
let L be with_infima antisymmetric RelStr;
let F be Subset of bool the carrier of L;
assume
A1: for X being Subset of L st X in F holds X is upper filtered;
reconsider F' = F as Subset-Family of L by SETFAM_1:def 7;
reconsider M = meet F' as Subset of L;
per cases;
suppose
A2: F = {};
M is filtered
proof
let x, y be Element of L;
assume x in M & y in M;
hence thesis by A2,SETFAM_1:def 1;
end;
hence thesis;
suppose
A3: F <> {};
M is filtered
proof
let x, y be Element of L such that
A4: x in M & y in M;
take z = x"/\"y;
for Y being set st Y in F holds z in Y
proof
let Y be set;
assume
A5: Y in F;
then reconsider X = Y as Subset of L;
A6: X is upper filtered by A1,A5;
x in X & y in X by A4,A5,SETFAM_1:def 1;
hence z in Y by A6,WAYBEL_0:41;
end;
hence z in M by A3,SETFAM_1:def 1;
thus z <= x & z <= y by YELLOW_0:23;
end;
hence thesis;
end;
theorem Th42:
for L being with_infima Poset
for I, J being Ideal of L
holds
I /\ J is Ideal of L
proof
let L be with_infima Poset;
let I, J be Ideal of L;
consider i being Element of I, j being Element of J;
set a = i"/\"j;
a <= i & a <= j by YELLOW_0:23;
then a in I & a in J by WAYBEL_0:def 19;
hence I /\ J is Ideal of L by WAYBEL_0:27,44,XBOOLE_0:def 3;
end;
definition
let L be non empty reflexive transitive RelStr;
cluster Ids L -> non empty;
correctness
proof
consider x being Element of L;
downarrow x in {Y where Y is Ideal of L: not contradiction};
hence Ids L is non empty by WAYBEL_0:def 23;
end;
end;
theorem Th43:
for L being non empty reflexive transitive RelStr
holds
(x is Element of InclPoset(Ids L) iff x is Ideal of L)
proof
let L be non empty reflexive transitive RelStr;
hereby
assume x is Element of InclPoset(Ids L);
then x in the carrier of InclPoset(Ids L);
then x in Ids L by YELLOW_1:1;
then x in {Y where Y is Ideal of L: not contradiction} by WAYBEL_0:def 23;
then consider J being Ideal of L such that
A1: J = x;
thus x is Ideal of L by A1;
end;
assume x is Ideal of L;
then x in {Y where Y is Ideal of L: not contradiction};
then x in Ids L by WAYBEL_0:def 23;
then x in the carrier of InclPoset(Ids L) by YELLOW_1:1;
hence x is Element of InclPoset(Ids L);
end;
theorem Th44:
for L being non empty reflexive transitive RelStr
for I being Element of InclPoset(Ids L)
holds
x in I implies x is Element of L
proof
let L be non empty reflexive transitive RelStr;
let I be Element of InclPoset(Ids L);
reconsider I'= I as non empty Subset of L by Th43;
assume x in I;
then reconsider x'= x as Element of I';
x' in the carrier of L;
hence x is Element of L;
end;
theorem
for L being with_infima Poset
for x, y being Element of InclPoset(Ids L)
holds
x "/\" y = x /\ y
proof
let L be with_infima Poset;
let x, y be Element of InclPoset(Ids L);
reconsider x'= x, y'= y as Ideal of L by Th43;
x' /\ y' is Ideal of L by Th42;
then x' /\ y' in {X where X is Ideal of L: not contradiction};
then x /\ y in Ids L by WAYBEL_0:def 23;
hence x /\ y = x "/\" y by YELLOW_1:9;
end;
definition
let L be with_infima Poset;
cluster InclPoset(Ids L) -> with_infima;
correctness
proof
for x, y be set st (x in Ids L & y in Ids L) holds x /\ y in Ids L
proof
let x, y be set;
assume
A1: x in Ids L & y in Ids L;
then x in
{X where X is Ideal of L: not contradiction} by WAYBEL_0:def 23;
then consider I being Ideal of L such that
A2: I = x;
y in {X where X is Ideal of L: not contradiction} by A1,WAYBEL_0:def 23
;
then consider J being Ideal of L such that
A3: J = y;
I /\ J is Ideal of L by Th42;
then I /\ J in {X where X is Ideal of L: not contradiction};
hence x /\ y in Ids L by A2,A3,WAYBEL_0:def 23;
end;
hence InclPoset(Ids L) is with_infima by YELLOW_1:12;
end;
end;
theorem Th46:
for L being with_suprema Poset
for x, y being Element of InclPoset(Ids L)
holds
ex Z being Subset of L st Z = {z where z is Element of L: z in x or z in
y or
ex a, b being Element of L st a in x & b in y & z = a "\/" b} &
ex_sup_of {x, y},InclPoset(Ids L) & x "\/" y = downarrow Z
proof
let L be with_suprema Poset;
set P = InclPoset(Ids L);
let x, y be Element of P;
reconsider x'= x, y'= y as Ideal of L by Th43;
defpred P[Element of L] means $1 in x or $1 in y or
ex a, b being Element of L st a in x & b in y & $1 = a "\/" b;
reconsider Z = {z where z is Element of L: P[z]}
as Subset of L from RelStrSubset;
take Z;
consider z being Element of x';
z in Z;
then reconsider Z as non empty Subset of L;
set DZ = downarrow Z;
for u, v being Element of L st u in Z & v in Z
ex z being Element of L st z in Z & u <= z & v <= z
proof
let u, v be Element of L such that
A1: u in Z and
A2: v in Z;
consider p being Element of L such that
A3: p = u and
A4: p in x or p in y or
ex a, b being Element of L st a in x & b in y & p = a "\/" b by A1;
consider q being Element of L such that
A5: q = v and
A6: q in x or q in y or
ex a, b being Element of L st a in x & b in y & q = a "\/" b by A2;
A7: for p, q being Element of L st p in x &
ex a, b being Element of L st a in x & b in y & q = a "\/" b
holds
ex z being Element of L st z in Z & p <= z & q <= z
proof
let p, q be Element of L such that
A8: p in x and
A9: ex a, b being Element of L st a in x & b in y & q = a "\/" b;
consider a, b being Element of L such that
A10: a in x and
A11: b in y and
A12: q = a "\/" b by A9;
reconsider c = a "\/" p as Element of L;
take z = c "\/" b;
c in x' by A8,A10,WAYBEL_0:40;
hence z in Z by A11;
A13: a <= c & p <= c & c <= z & b <= z by YELLOW_0:22;
then a <= z & b <= z by ORDERS_1:26;
hence p <= z & q <= z by A12,A13,ORDERS_1:26,YELLOW_0:22;
end;
A14: for p, q being Element of L st p in y &
ex a, b being Element of L st a in x & b in y & q = a "\/" b
holds
ex z being Element of L st z in Z & p <= z & q <= z
proof
let p, q be Element of L such that
A15: p in y and
A16: ex a, b being Element of L st a in x & b in y & q = a "\/" b;
consider a, b being Element of L such that
A17: a in x and
A18: b in y and
A19: q = a "\/" b by A16;
reconsider c = b "\/" p as Element of L;
take z = a "\/" c;
c in y' by A15,A18,WAYBEL_0:40;
hence z in Z by A17;
A20: b <= c & p <= c & a <= z & c <= z by YELLOW_0:22;
then a <= z & b <= z by ORDERS_1:26;
hence p <= z & q <= z by A19,A20,ORDERS_1:26,YELLOW_0:22;
end;
per cases by A4,A6;
suppose p in x & q in x;
then consider z being Element of L such that
A21: z in x' & p <= z & q <= z by WAYBEL_0:def 1;
take z;
thus thesis by A3,A5,A21;
suppose
A22: p in x & q in y;
take p "\/" q;
thus thesis by A3,A5,A22,YELLOW_0:22;
suppose p in x & ex a, b being Element of L st
a in x & b in y & q = a "\/" b;
then consider z being Element of L such that
A23: z in Z & p <= z & q <= z by A7;
take z;
thus thesis by A3,A5,A23;
suppose
A24: p in y & q in x;
take q "\/" p;
thus thesis by A3,A5,A24,YELLOW_0:22;
suppose p in y & q in y;
then consider z being Element of L such that
A25: z in y' & p <= z & q <= z by WAYBEL_0:def 1;
take z;
thus thesis by A3,A5,A25;
suppose p in y & ex a, b being Element of L st
a in x & b in y & q = a "\/" b;
then consider z being Element of L such that
A26: z in Z & p <= z & q <= z by A14;
take z;
thus thesis by A3,A5,A26;
suppose q in x & ex a, b being Element of L st
a in x & b in y & p = a "\/" b;
then consider z being Element of L such that
A27: z in Z & q <= z & p <= z by A7;
take z;
thus thesis by A3,A5,A27;
suppose q in y & ex a, b being Element of L st
a in x & b in y & p = a "\/" b;
then consider z being Element of L such that
A28: z in Z & q <= z & p <= z by A14;
take z;
thus thesis by A3,A5,A28;
suppose (ex a, b being Element of L st a in x & b in y & p = a "\/" b) &
(ex a, b being Element of L st a in x & b in y & q = a "\/" b);
then consider a, b, c, d being Element of L such that
A29: a in x & b in y & p = a "\/" b and
A30: c in x & d in y & q = c "\/" d;
reconsider ac = a "\/" c, bd = b "\/" d as Element of L;
take z = ac "\/" bd;
ac in x' & bd in y' by A29,A30,WAYBEL_0:40;
hence z in Z;
A31: a <= ac & c <= ac & b <= bd & d <= bd & ac <= z & bd <= z
by YELLOW_0:22;
then a <= z & b <= z by ORDERS_1:26;
hence u <= z by A3,A29,YELLOW_0:22;
c <= z & d <= z by A31,ORDERS_1:26;
hence v <= z by A5,A30,YELLOW_0:22;
end;
then Z is directed by WAYBEL_0:def 1;
then DZ is Ideal of L by WAYBEL_0:30;
then reconsider DZ as Element of P by Th43;
x c= DZ
proof
let a be set;
assume
A32: a in x;
then reconsider a'= a as Element of L by Th44;
a' in Z & Z c= DZ by A32,WAYBEL_0:16;
hence a in DZ;
end;
then A33: x <= DZ by YELLOW_1:3;
y c= DZ
proof
let a be set;
assume
A34: a in y;
then reconsider a'= a as Element of L by Th44;
a' in Z & Z c= DZ by A34,WAYBEL_0:16;
hence a in DZ;
end;
then A35: y <= DZ by YELLOW_1:3;
for d being Element of P st d >= x & d >= y holds DZ <= d
proof
let d be Element of P;
assume x <= d & y <= d;
then A36: x c= d & y c= d by YELLOW_1:3;
DZ c= d
proof
let p be set;
assume p in DZ;
then p in {q where q is Element of L:
ex u being Element of L st q <= u & u in Z} by WAYBEL_0:14;
then consider p' being Element of L such that
A37: p' = p and
A38: ex u being Element of L st p' <= u & u in Z;
consider u being Element of L such that
A39: p' <= u and
A40: u in Z by A38;
consider z being Element of L such that
A41: z = u and
A42: z in x or z in y or
ex a, b being Element of L st a in x & b in y & z = a "\/" b by A40;
per cases by A42;
suppose z in x;
then p in x' by A37,A39,A41,WAYBEL_0:def 19;
hence p in d by A36;
suppose z in y;
then p in y' by A37,A39,A41,WAYBEL_0:def 19;
hence p in d by A36;
suppose ex a, b being Element of L st a in x & b in y & z = a "\/"
b;
then consider a, b being Element of L such that
A43: a in x and
A44: b in y and
A45: z = a "\/" b;
reconsider d'= d as Ideal of L by Th43;
u in d' by A36,A41,A43,A44,A45,WAYBEL_0:40;
hence p in d by A37,A39,WAYBEL_0:def 19;
end;
hence DZ <= d by YELLOW_1:3;
end;
hence thesis by A33,A35,YELLOW_0:18;
end;
definition
let L be with_suprema Poset;
cluster InclPoset(Ids L) -> with_suprema;
correctness
proof
set P = InclPoset(Ids L);
for x, y being Element of P
ex z being Element of P st x <= z & y <= z &
for z' being Element of P st x <= z' & y <= z' holds z <= z'
proof
let x, y be Element of P;
consider Z being Subset of L such that
Z = {z where z is Element of L: z in x or z in y or
ex a, b being Element of L st a in x & b in y & z = a "\/" b} and
A1: ex_sup_of {x, y},InclPoset(Ids L) and
x "\/" y = downarrow Z by Th46;
take x "\/" y;
thus thesis by A1,YELLOW_0:18;
end;
hence thesis by LATTICE3:def 10;
end;
end;
theorem Th47:
for L being lower-bounded sup-Semilattice
for X being non empty Subset of Ids L
holds
meet X is Ideal of L
proof
let L be lower-bounded sup-Semilattice;
let X be non empty Subset of Ids L;
A1: for J being set st J in X holds J is Ideal of L
proof
let J be set;
assume J in X;
then J in Ids L;
then J in {Y where Y is Ideal of L: not contradiction} by WAYBEL_0:def 23;
then ex Y being Ideal of L st Y = J;
hence thesis;
end;
X c= bool the carrier of L
proof
let x;
assume x in X;
then x is Ideal of L by A1;
hence x in bool the carrier of L;
end;
then reconsider F = X as Subset of bool the carrier of L;
for J being Subset of L st J in F holds J is lower by A1;
then reconsider I = meet X as lower Subset of L by Th38;
for J being set st J in X holds Bottom L in J
proof
let J be set;
assume
J in X;
then reconsider J'= J as Ideal of L by A1;
consider j being Element of J';
Bottom L <= j by YELLOW_0:44;
hence Bottom L in J by WAYBEL_0:def 19;
end;
then reconsider I as non empty lower Subset of L by SETFAM_1:def 1;
for J being Subset of L st J in F holds J is lower directed by A1;
then I is Ideal of L by Th40;
hence thesis;
end;
theorem Th48:
for L being lower-bounded sup-Semilattice
for A being non empty Subset of InclPoset(Ids L)
holds
ex_inf_of A,InclPoset(Ids L) & inf A = meet A
proof
let L be lower-bounded sup-Semilattice;
let A be non empty Subset of InclPoset(Ids L);
set P = InclPoset(Ids L);
A c= the carrier of InclPoset Ids L; then
A c= Ids L by YELLOW_1:1; then
reconsider A'= A as non empty Subset of Ids L;
meet A' is Ideal of L by Th47;
then reconsider I = meet A as Element of P by Th43;
A1: I is_<=_than A
proof
let y be Element of P;
assume
A2: y in A;
I c= y
proof
let x be set;
assume x in I;
hence x in y by A2,SETFAM_1:def 1;
end;
hence I <= y by YELLOW_1:3;
end;
for b being Element of P st b is_<=_than A holds I >= b
proof
let b be Element of P;
assume
A3: A is_>=_than b;
A4: for J being set st J in A holds b c= J
proof
let J be set;
assume
A5: J in A;
then J is Element of A;
then reconsider y = J as Element of P;
b <= y by A3,A5,LATTICE3:def 8;
hence b c= J by YELLOW_1:3;
end;
b c= I
proof
let c be set;
assume
A6: c in b;
for J being set st J in A holds c in J
proof
let J be set;
assume J in A;
then b c= J by A4;
hence c in J by A6;
end;
hence c in I by SETFAM_1:def 1;
end;
hence I >= b by YELLOW_1:3;
end;
hence thesis by A1,YELLOW_0:31;
end;
theorem Th49:
for L being with_suprema Poset
holds
ex_inf_of {},InclPoset(Ids L) & "/\"({}, InclPoset(Ids L)) = [#]L
proof
let L be with_suprema Poset;
set P = InclPoset(Ids L);
reconsider I = [#]L as Element of P by Th43;
A1: I is_<=_than {}
proof
let y be Element of P;
assume y in {};
hence thesis;
end;
for b being Element of P st b is_<=_than {} holds I >= b
proof
let b be Element of P;
assume {} is_>=_than b;
reconsider b'= b as Ideal of L by Th43;
b' c= the carrier of L;
then b' c= [#]L by PRE_TOPC:12;
hence I >= b by YELLOW_1:3;
end;
hence thesis by A1,YELLOW_0:31;
end;
theorem Th50:
for L being lower-bounded sup-Semilattice
holds
InclPoset(Ids L) is complete
proof
let L be lower-bounded sup-Semilattice;
set P = InclPoset(Ids L);
for A being Subset of P holds ex_inf_of A,InclPoset(Ids L)
proof
let A be Subset of P;
per cases;
suppose A = {};
hence thesis by Th49;
suppose A <> {};
hence thesis by Th48;
end;
hence InclPoset(Ids L) is complete by Th30;
end;
definition
let L be lower-bounded sup-Semilattice;
cluster InclPoset(Ids L) -> complete;
correctness by Th50;
end;
begin :: Special Maps
definition
let L be non empty Poset;
func SupMap L -> map of InclPoset(Ids L), L means
:Def3:
for I being Ideal of L holds it.I = sup I;
existence
proof
set P = InclPoset(Ids L);
deffunc F(set) = "\/"($1, L);
A1: for I being set st I in the carrier of P holds
F(I) in the carrier of L;
ex f being Function of the carrier of P, the carrier of L st
for I being set st I in the carrier of P holds f.I = F(I)
from Lambda1(A1);
then consider f being Function of the carrier of P, the carrier of L such
that
A2: for I being set st I in the carrier of P holds f.I = "\/"(I, L);
reconsider f as map of P, L;
take f;
for I being Ideal of L holds f.I = sup I
proof
let I be Ideal of L;
I in {X where X is Ideal of L: not contradiction};
then I in the carrier of RelStr(#Ids L, RelIncl Ids L#)
by WAYBEL_0:def 23;
then I in the carrier of P by YELLOW_1:def 1;
hence f.I = sup I by A2;
end;
hence thesis;
end;
uniqueness
proof
let f, g be map of InclPoset(Ids L), L;
assume that
A3: for I being Ideal of L holds f.I = sup I and
A4: for I being Ideal of L holds g.I = sup I;
set P = InclPoset(Ids L);
A5: dom f = the carrier of P & dom g = the carrier of P by FUNCT_2:def 1;
A6: the carrier of P = the carrier of RelStr(#Ids L, RelIncl(Ids L)#)
by YELLOW_1:def 1
.= Ids L;
now
let x be set;
assume x in the carrier of P;
then x in {X where X is Ideal of L: not contradiction} by A6,WAYBEL_0:def
23;
then ex I being Ideal of L st x = I;
then reconsider I = x as Ideal of L;
f.I = sup I & g.I = sup I by A3,A4;
hence f.x = g.x;
end;
hence f = g by A5,FUNCT_1:9;
end;
end;
theorem Th51:
for L being non empty Poset
holds
dom SupMap L = Ids L & rng SupMap L is Subset of L
proof
let L be non empty Poset;
set P = InclPoset(Ids L);
thus dom(SupMap L) = the carrier of P by FUNCT_2:def 1
.= the carrier of RelStr(#Ids L, RelIncl(Ids L)#) by YELLOW_1:def 1
.= Ids L;
thus rng(SupMap L) is Subset of L;
end;
theorem
for L being non empty Poset
holds
x in dom SupMap L iff x is Ideal of L
proof
let L be non empty Poset;
hereby
assume x in dom SupMap L;
then x in Ids L by Th51;
then x in {X where X is Ideal of L: not contradiction} by WAYBEL_0:def 23;
then ex I being Ideal of L st x = I;
hence x is Ideal of L;
end;
assume x is Ideal of L;
then x in {X where X is Ideal of L: not contradiction};
then x in Ids L by WAYBEL_0:def 23;
hence x in dom(SupMap L) by Th51;
end;
theorem Th53:
for L being up-complete (non empty Poset) holds SupMap L is monotone
proof
let L be up-complete (non empty Poset);
set P = InclPoset Ids L;
set f = SupMap L;
for x, y being Element of P st x <= y
for a, b being Element of L st a = f.x & b = f.y holds a <= b
proof
let x, y be Element of P such that
A1: x <= y;
let a, b be Element of L such that
A2: a = f.x & b = f.y;
reconsider I = x, J = y as Ideal of L by Th43;
A3: ex_sup_of I,L & ex_sup_of J,L by WAYBEL_0:75;
A4: I c= J by A1,YELLOW_1:3;
f.x = sup I & f.y = sup J by Def3;
hence thesis by A2,A3,A4,YELLOW_0:34;
end;
hence thesis by ORDERS_3:def 5;
end;
definition
let L be up-complete (non empty Poset);
cluster SupMap L -> monotone;
correctness by Th53;
end;
definition
let L be non empty Poset;
func IdsMap L -> map of L, InclPoset(Ids L) means
:Def4:
for x being Element of L holds it.x = downarrow x;
existence
proof
deffunc F(Element of L) = downarrow $1;
A1: for x being Element of L holds F(x) is Element of InclPoset(Ids L)
by Th43;
thus ex m being map of L, InclPoset Ids L st
for x being Element of L holds m.x = F(x) from KappaMD(A1);
end;
uniqueness
proof let f1,f2 be map of L,InclPoset(Ids L) such that
A2: for x being Element of L holds f1.x = downarrow x and
A3: for x being Element of L holds f2.x = downarrow x;
now let x be Element of L;
thus f1.x = downarrow x by A2
.= f2.x by A3;
end;
hence thesis by Th9;
end;
end;
theorem Th54:
for L being non empty Poset holds IdsMap L is monotone
proof
let L be non empty Poset;
let x1,x2 be Element of L such that
A1: x1 <= x2;
let a, b be Element of InclPoset(Ids L) such that
A2: a = (IdsMap L).x1 & b = (IdsMap L).x2;
downarrow x1 c= downarrow x2 by A1,WAYBEL_0:21;
then (IdsMap L).x1 c= downarrow x2 by Def4;
then (IdsMap L).x1 c= (IdsMap L).x2 by Def4;
hence a <= b by A2,YELLOW_1:3;
end;
definition
let L be non empty Poset;
cluster IdsMap L -> monotone;
correctness by Th54;
end;
begin :: A Family of Elements in a Lattice
definition
let L be non empty RelStr;
let F be Relation;
func \\/(F, L) -> Element of L equals
:Def5:
"\/"(rng F, L);
coherence;
func //\(F, L) -> Element of L equals
:Def6:
"/\"(rng F, L);
coherence;
end;
definition
let J be set, L be non empty RelStr;
let F be Function of J, the carrier of L;
redefine func \\/(F, L);
synonym Sup F;
redefine func //\(F, L);
synonym Inf F;
end;
definition
let J be non empty set, S be non empty 1-sorted;
let F be Function of J, the carrier of S;
let j be Element of J;
redefine func F.j -> Element of S;
coherence
proof
F.j in the carrier of S;
hence thesis;
end;
end;
definition
let J be set, S be non empty 1-sorted;
let F be Function of J, the carrier of S;
redefine func rng F -> Subset of S;
coherence by RELSET_1:12;
end;
reserve J for non empty set,
j for Element of J;
theorem
for F being Function of J, the carrier of L
holds
F.j <= Sup F & Inf F <= F.j
proof
let F be Function of J, the carrier of L;
F.j in rng F by FUNCT_2:6;
then F.j <= "\/"(rng F, L) & "/\"(rng F, L) <= F.j by Th24;
hence F.j <= Sup F & Inf F <= F.j by Def5,Def6;
end;
theorem
for F being Function of J, the carrier of L
holds
(for j holds F.j <= a) implies Sup F <= a
proof
let F be Function of J, the carrier of L;
assume
A1: for j holds F.j <= a;
now
let c be Element of L;
assume c in rng F;
then consider j being set such that
A2: j in dom F and
A3: c = F.j by FUNCT_1:def 5;
reconsider j as Element of J by A2,FUNCT_2:def 1;
c = F.j by A3;
hence c <= a by A1;
end;
then rng F is_<=_than a by LATTICE3:def 9;
then "\/"(rng F, L) <= a by YELLOW_0:32;
hence thesis by Def5;
end;
theorem
for F being Function of J, the carrier of L
holds
(for j holds a <= F.j) implies a <= Inf F
proof
let F be Function of J, the carrier of L;
assume
A1: for j holds a <= F.j;
now
let c be Element of L;
assume c in rng F;
then consider j being set such that
A2: j in dom F and
A3: c = F.j by FUNCT_1:def 5;
reconsider j as Element of J by A2,FUNCT_2:def 1;
c = F.j by A3;
hence a <= c by A1;
end;
then a is_<=_than rng F by LATTICE3:def 8;
then a <= "/\"(rng F, L) by YELLOW_0:33;
hence thesis by Def6;
end;