:: A theory of partitions, { I }
:: by Shunichi Kobayashi and Kui Jia
::
:: Received October 5, 1998
:: Copyright (c) 1998-2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, XBOOLE_0, EQREL_1, SUBSET_1, TARSKI, SETFAM_1, FUNCT_1,
RELAT_1, ZFMISC_1, FINSEQ_1, XXREAL_0, ARYTM_3, CARD_1, ORDINAL4,
PARTIT1, NAT_1;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0,
NAT_1, RELAT_1, FUNCT_1, FUNCT_2, SETFAM_1, FINSEQ_1, FINSEQ_4, EQREL_1,
XXREAL_0;
constructors SETFAM_1, FUNCT_2, XXREAL_0, XREAL_0, NAT_1, MEMBERED, EQREL_1,
FINSEQ_4, RELSET_1, DOMAIN_1, NUMBERS;
registrations XBOOLE_0, SUBSET_1, RELSET_1, PARTFUN1, MEMBERED, EQREL_1,
NAT_1, XXREAL_0;
requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
definitions TARSKI, RELAT_1;
equalities EQREL_1;
expansions TARSKI, RELAT_1;
theorems TARSKI, FUNCT_1, SETFAM_1, EQREL_1, ZFMISC_1, RELAT_1, NAT_1,
FINSEQ_1, FINSEQ_2, FUNCT_2, SUBSET_1, RELSET_1, XBOOLE_0, XBOOLE_1,
XREAL_1, XXREAL_0;
schemes EQREL_1, NAT_1, CLASSES1, DOMAIN_1, XFAMILY;
begin
::Chap. 1 Preliminaries
reserve Y,Z for non empty set;
reserve PA,PB for a_partition of Y;
reserve A,B for Subset of Y;
reserve i,j,k for Nat;
reserve x,y,z,x1,x2,y1,z0,X,V,a,b,d,t,SFX,SFY for set;
theorem Th1:
X in PA & V in PA & X c= V implies X=V
proof
assume that
A1: X in PA and
A2: V in PA and
A3: X c= V;
A4: X=V or X misses V by A1,A2,EQREL_1:def 4;
set x = the Element of X;
X<>{} by A1,EQREL_1:def 4;
then x in X;
hence thesis by A3,A4,XBOOLE_0:3;
end;
notation
let SFX,SFY;
synonym SFX '<' SFY for SFX is_finer_than SFY;
synonym SFY '>' SFX for SFX is_finer_than SFY;
end;
theorem Th2:
union (SFX \ {{}}) = union SFX
proof
A1: union (SFX \ {{}}) c= (union SFX)
proof
let y be object;
assume y in union (SFX \ {{}});
then ex z being set st y in z & z in (SFX \{{} }) by TARSKI:def 4;
hence thesis by TARSKI:def 4;
end;
union SFX c= union (SFX \ {{}})
proof
let y be object;
assume y in union SFX;
then consider X being set such that
A2: y in X and
A3: X in SFX by TARSKI:def 4;
not X in {{}} by A2,TARSKI:def 1;
then X in SFX \ {{}} by A3,XBOOLE_0:def 5;
hence thesis by A2,TARSKI:def 4;
end;
hence thesis by A1,XBOOLE_0:def 10;
end;
theorem Th3:
for PA,PB being a_partition of Y st PA '>' PB & PB '>' PA holds PB c= PA
proof
let PA,PB be a_partition of Y;
assume that
A1: PA '>' PB and
A2: PB '>' PA;
let x be object;
reconsider xx=x as set by TARSKI:1;
assume
A3: x in PB;
then consider V such that
A4: V in PA and
A5: xx c= V by A1,SETFAM_1:def 2;
consider W being set such that
A6: W in PB and
A7: V c= W by A2,A4,SETFAM_1:def 2;
x=W by A3,A5,A6,A7,Th1,XBOOLE_1:1;
hence thesis by A4,A5,A7,XBOOLE_0:def 10;
end;
theorem Th4:
for PA,PB being a_partition of Y st PA '>' PB & PB '>' PA holds PA = PB
proof
let PA,PB be a_partition of Y;
assume PA '>' PB & PB '>' PA;
then PB c= PA & PA c= PB by Th3;
hence thesis by XBOOLE_0:def 10;
end;
theorem Th5:
for PA,PB being a_partition of Y st PA '>' PB holds PA is_coarser_than PB
proof
let PA,PB be a_partition of Y;
assume
A1: PA '>' PB;
for x being set st x in PA ex y being set st y in PB & y c= x
proof
let x be set;
assume
A2: x in PA;
then A3: x<>{} by EQREL_1:def 4;
set u = the Element of x;
A4: u in x by A3;
union PB = Y by EQREL_1:def 4;
then consider y being set such that
A5: u in y and
A6: y in PB by A2,A4,TARSKI:def 4;
consider z being set such that
A7: z in PA and
A8: y c= z by A1,A6,SETFAM_1:def 2;
x=z or x misses z by A2,A7,EQREL_1:def 4;
hence thesis by A3,A5,A6,A8,XBOOLE_0:3;
end;
hence thesis by SETFAM_1:def 3;
end;
definition
let Y;
let PA be a_partition of Y,b be set;
pred b is_a_dependent_set_of PA means
ex B being set st B c= PA & B<>{} & b = union B;
end;
definition
let Y;
let PA,PB be a_partition of Y,b be set;
pred b is_min_depend PA,PB means
b is_a_dependent_set_of PA &
b is_a_dependent_set_of PB & for d being set st d c= b
& d is_a_dependent_set_of PA & d is_a_dependent_set_of PB holds d=b;
end;
theorem Th6:
for PA,PB being a_partition of Y
st PA '>' PB holds for b being set st b in PA holds
b is_a_dependent_set_of PB
proof
let PA,PB be a_partition of Y;
assume
A1: PA '>' PB;
A2: union PB = Y by EQREL_1:def 4;
A3: PA is_coarser_than PB by A1,Th5;
for b being set st b in PA holds b is_a_dependent_set_of PB
proof
let b be set;
assume
A4: b in PA;
set B0={x8 where x8 is Subset of Y: x8 in PB & x8 c= b};
consider xb be set such that
A5: xb in PB & xb c= b by A3,A4,SETFAM_1:def 3;
A6: xb in B0 by A5;
for z being object holds z in B0 implies z in PB
proof let z be object;
assume z in B0;
then ex x8 being Subset of Y st x8=z & x8 in PB & x8 c= b;
hence thesis;
end;
then A7: B0 c= PB;
for z being object holds z in b implies z in union B0
proof let z be object;
assume
A8: z in b;
then consider x1 such that
A9: z in x1 and
A10: x1 in PB by A2,A4,TARSKI:def 4;
consider y1 such that
A11: y1 in PA and
A12: x1 c= y1 by A1,A10,SETFAM_1:def 2;
b = y1 or b misses y1 by A4,A11,EQREL_1:def 4;
then x1 in B0 by A8,A9,A10,A12,XBOOLE_0:3;
hence thesis by A9,TARSKI:def 4;
end;
then A13: b c= union B0;
for z being object holds z in union B0 implies z in b
proof let z be object;
assume z in union B0;
then consider y such that
A14: z in y and
A15: y in B0 by TARSKI:def 4;
ex x8 being Subset of Y st x8=y & x8 in PB & x8 c= b by A15;
hence thesis by A14;
end;
then union B0 c= b;
then b = union B0 by A13,XBOOLE_0:def 10;
hence thesis by A6,A7;
end;
hence thesis;
end;
theorem Th7:
for PA being a_partition of Y holds Y is_a_dependent_set_of PA
proof
let PA be a_partition of Y;
A1: {Y} is Subset-Family of Y by ZFMISC_1:68;
A2: union {Y} = Y by ZFMISC_1:25;
for
A st A in {Y} holds A<>{} & for B st B in {Y} holds A = B or A misses B
proof
let A;
assume
A3: A in {Y};
then A4: A=Y by TARSKI:def 1;
thus A<>{} by A3,TARSKI:def 1;
let B;
assume B in {Y};
hence thesis by A4,TARSKI:def 1;
end;
then A5: {Y} is a_partition of Y by A1,A2,EQREL_1:def 4;
for a being set st a in PA ex b being set st b in {Y} & a c= b
proof
let a be set;
assume
A6: a in PA;
then A7: a<>{} by EQREL_1:def 4;
set x = the Element of a;
x in Y by A6,A7,TARSKI:def 3;
then consider b being set such that
x in b and
A8: b in {Y} by A2,TARSKI:def 4;
b = Y by A8,TARSKI:def 1;
hence thesis by A6,A8;
end;
then A9: {Y} '>' PA by SETFAM_1:def 2;
Y in {Y} by TARSKI:def 1;
hence thesis by A5,A9,Th6;
end;
theorem Th8:
for F being Subset-Family of Y st (Intersect(F)<>{} &
for X st X in F holds X is_a_dependent_set_of PA)
holds Intersect(F) is_a_dependent_set_of PA
proof
let F be Subset-Family of Y;
per cases;
suppose
F={};
then Intersect(F)=Y by SETFAM_1:def 9;
hence thesis by Th7;
end;
suppose
A1: F<>{};
then reconsider F9 = F as non empty Subset-Family of Y;
assume that
A2: Intersect(F)<>{} and
A3: for X st X in F holds X is_a_dependent_set_of PA;
defpred P[object,object] means
ex A being set st A = $2 & A c= PA & $2<>{} & $1=union A;
A4: for X being object st X in F holds ex B being object st P[X,B]
proof let x be object;
reconsider X = x as set by TARSKI:1;
assume
x in F;
then X is_a_dependent_set_of PA by A3;
then ex B being set st B c= PA & B<>{} & X = union B;
then ex B being object st P[X,B];
hence thesis;
end;
consider f being Function such that
A5: dom f = F &
for X being object st X in F holds P[X,f.X] from CLASSES1:sch 1(A4);
rng f c= bool bool Y
proof
let y be object;
reconsider yy=y as set by TARSKI:1;
assume y in rng f;
then consider x being object such that
A6: x in dom f and
A7: y = f.x by FUNCT_1:def 3;
P[x,f.x] by A5,A6;
then f.x c= PA;
then yy c= bool Y by A7,XBOOLE_1:1;
hence thesis;
end;
then reconsider f as Function of F9, bool bool Y
by A5,FUNCT_2:def 1,RELSET_1:4;
defpred P[set] means $1 in F;
deffunc S(Element of F9) = f.$1;
reconsider T={S(X) where X is Element of F9:P[X]}
as Subset-Family of bool Y from DOMAIN_1:sch 8;
reconsider T as Subset-Family of bool Y;
take B=Intersect(T);
thus B c= PA
proof
let x be object;
assume
A8: x in B;
consider X being object such that
A9: X in F by A1,XBOOLE_0:def 1;
f.X in T by A9;
then A10: x in f.X by A8,SETFAM_1:43;
P[X,f.X] by A5,A9;
then f.X c= PA;
hence thesis by A10;
end;
thus B<>{}
proof
consider X being object such that
A11: X in F by A1,XBOOLE_0:def 1;
A12: f.X in T by A11;
consider A being object such that
A13: A in Intersect(F) by A2,XBOOLE_0:def 1;
reconsider A as Element of Y by A13;
set AA = EqClass(A,PA);
A14: A in meet F by A1,A13,SETFAM_1:def 9;
for X st X in T holds AA in X
proof
let X;
assume X in T;
then consider z being Element of F9 such that
A15: X=f.z and z in F;
A16: P[z,f.z] by A5;
then
A17: f.z c= PA;
z = union (f.z) & A in z by A14,SETFAM_1:def 1,A16;
then ex A0 being set st A in A0 & A0 in f.z by TARSKI:def 4;
hence thesis by A15,A17,EQREL_1:def 6;
end;
then EqClass(A,PA) in meet T by A12,SETFAM_1:def 1;
hence thesis by SETFAM_1:def 9;
end;
A18: Intersect(F) c= union B
proof
let x be object;
assume
A19: x in Intersect(F);
then A20: x in meet F by A1,SETFAM_1:def 9;
reconsider x as Element of Y by A19;
reconsider PA as a_partition of Y;
set A = EqClass(x,PA);
consider X being object such that
A21: X in F by A1,XBOOLE_0:def 1;
A22: f.X in T by A21;
for X st X in T holds A in X
proof
let X;
assume X in T;
then consider z being Element of F9 such that
A23: X=f.z and z in F;
A24: P[z,f.z] by A5;
then
A25: f.z c= PA;
z = union (f.z) by A24;
then x in union (f.z) by A20,SETFAM_1:def 1;
then ex A0 being set st x in A0 & A0 in f.z by TARSKI:def 4;
hence thesis by A23,A25,EQREL_1:def 6;
end;
then A in meet T by A22,SETFAM_1:def 1;
then x
in A & A in Intersect(T) by A22,EQREL_1:def 6,SETFAM_1:def 9;
hence thesis by TARSKI:def 4;
end;
union B c= Intersect(F)
proof
let x be object;
assume
A26: x in union B;
consider X being object such that
A27: X in F by A1,XBOOLE_0:def 1;
A28: f.X in T by A27;
consider y being set such that
A29: x in y and
A30: y in B by A26,TARSKI:def 4;
A31: y in meet T by A28,A30,SETFAM_1:def 9;
for X st X in F holds x in X
proof
let X;
assume
A32: X in F;
then f.X in T;
then A33: y in f.X by A31,SETFAM_1:def 1;
P[X,f.X] by A5,A32;
then X = union (f.X);
hence thesis by A29,A33,TARSKI:def 4;
end;
then x in meet F by A1,SETFAM_1:def 1;
hence thesis by A1,SETFAM_1:def 9;
end;
hence Intersect(F) = union B by A18,XBOOLE_0:def 10;
end;
end;
theorem Th9:
for X0,X1 being Subset of Y st X0 is_a_dependent_set_of PA &
X1 is_a_dependent_set_of PA & X0 meets X1
holds X0 /\ X1 is_a_dependent_set_of PA
proof
let X0,X1 be Subset of Y;
assume that
A1: X0 is_a_dependent_set_of PA and
A2: X1 is_a_dependent_set_of PA and
A3: X0 meets X1;
consider A being set such that
A4: A c= PA and A<>{} and
A5: X0 = union A by A1;
consider B being set such that
A6: B c= PA and B<>{} and
A7: X1 = union B by A2;
A8: X0 /\ X1 c= union (A /\ B)
proof
let x be object;
assume
A9: x in (X0 /\ X1);
then A10: x in X0 by XBOOLE_0:def 4;
A11: x in X1 by A9,XBOOLE_0:def 4;
consider y being set such that
A12: x in y and
A13: y in A by A5,A10,TARSKI:def 4;
consider z being set such that
A14: x in z and
A15: z in B by A7,A11,TARSKI:def 4;
A16: y in PA by A4,A13;
A17: z in PA by A6,A15;
y meets z by A12,A14,XBOOLE_0:3;
then y = z by A16,A17,EQREL_1:def 4;
then y in A /\ B by A13,A15,XBOOLE_0:def 4;
hence thesis by A12,TARSKI:def 4;
end;
union (A /\ B) c= X0 /\ X1
proof
let x be object;
assume x in union (A /\ B);
then consider y being set such that
A18: x in y and
A19: y in (A /\ B) by TARSKI:def 4;
A20: y in A by A19,XBOOLE_0:def 4;
A21: y in B by A19,XBOOLE_0:def 4;
A22: x in X0 by A5,A18,A20,TARSKI:def 4;
x in X1 by A7,A18,A21,TARSKI:def 4;
hence thesis by A22,XBOOLE_0:def 4;
end;
then A23: X0 /\ X1 = union (A /\ B) by A8,XBOOLE_0:def 10;
A24: A \/ B c= PA by A4,A6,XBOOLE_1:8;
A /\ B c= A & A c= A \/ B by XBOOLE_1:7,17;
then A /\ B c= A \/ B;
then A25: A /\ B c= PA by A24;
now
assume
A26: A /\ B={};
ex y being object st y in X0 & y in X1 by A3,XBOOLE_0:3;
hence contradiction by A8,A26,XBOOLE_0:def 4,ZFMISC_1:2;
end;
hence thesis by A23,A25;
end;
theorem Th10:
for X being Subset of Y holds X is_a_dependent_set_of PA & X<>Y
implies X` is_a_dependent_set_of PA
proof
let X be Subset of Y;
assume that
A1: X is_a_dependent_set_of PA and
A2: X<>Y;
consider B being set such that
A3: B c= PA and B<>{} and
A4: X=union B by A1;
take PA \ B;
A5: union PA = Y by EQREL_1:def 4;
then A6: X` = union PA \ union B by A4,SUBSET_1:def 4;
reconsider B as Subset of PA by A3;
now
assume PA \ B={};
then PA c= B by XBOOLE_1:37;
hence contradiction by A2,A4,A5,XBOOLE_0:def 10;
end;
hence thesis by A6,EQREL_1:43,XBOOLE_1:36;
end;
theorem Th11:
for y being Element of Y
ex X being Subset of Y st y in X & X is_min_depend PA,PB
proof
let y be Element of Y;
A1: union PA = Y by EQREL_1:def 4;
A2: for A being set st A in PA holds A<>{} & for B being set st B in PA
holds A=B or A misses B by EQREL_1:def 4;
A3: Y is_a_dependent_set_of PA & Y is_a_dependent_set_of PB by Th7;
defpred P[set] means
y in $1 & $1 is_a_dependent_set_of PA & $1 is_a_dependent_set_of PB;
reconsider XX={X where X is Subset of Y:P[X]}
as Subset-Family of Y from DOMAIN_1:sch 7;
reconsider XX as Subset-Family of Y;
Y c= Y;
then A4: Y in XX by A3;
for X1 be set st X1 in XX holds y in X1
proof
let X1 be set;
assume X1 in XX;
then ex X be Subset of Y st X=X1 & y in X & X
is_a_dependent_set_of PA & X is_a_dependent_set_of PB;
hence thesis;
end;
then A5: y in meet XX by A4,SETFAM_1:def 1;
then A6: Intersect(XX)<>{} by SETFAM_1:def 9;
take Intersect(XX);
for X1 be set st X1 in XX holds X1 is_a_dependent_set_of PA
proof
let X1 be set;
assume X1 in XX;
then ex X be Subset of Y st X=X1 & y in X & X is_a_dependent_set_of PA &
X is_a_dependent_set_of PB;
hence thesis;
end;
then A7: Intersect(XX) is_a_dependent_set_of PA by A6,Th8;
for X1 be set st X1 in XX holds X1 is_a_dependent_set_of PB
proof
let X1 be set;
assume X1 in XX;
then ex X be Subset of Y st X=X1 & y in X & X is_a_dependent_set_of PA &
X is_a_dependent_set_of PB;
hence thesis;
end;
then A8: Intersect(XX) is_a_dependent_set_of PB by A6,Th8;
for d being set st d c= Intersect(XX) & d is_a_dependent_set_of PA &
d is_a_dependent_set_of PB holds d=Intersect(XX)
proof
let d be set;
assume that
A9: d c= Intersect(XX) and
A10: d is_a_dependent_set_of PA and
A11: d is_a_dependent_set_of PB;
consider Ad being set such that
A12: Ad c= PA and
A13: Ad<>{} and
A14: d = union Ad by A10;
A15: d c= Y by A1,A12,A14,ZFMISC_1:77;
per cases;
suppose
y in d;
then A16: d in XX by A10,A11,A15;
Intersect(XX) c= d
proof
let X1 be object;
assume X1 in Intersect(XX);
then X1 in meet XX by A4,SETFAM_1:def 9;
hence thesis by A16,SETFAM_1:def 1;
end;
hence thesis by A9,XBOOLE_0:def 10;
end;
suppose
A17: not y in d;
reconsider d as Subset of Y by A1,A12,A14,ZFMISC_1:77;
d` = Y \ d by SUBSET_1:def 4;
then A18: y in d` by A17,XBOOLE_0:def 5;
d misses d` by SUBSET_1:24;
then A19: d /\ d` = {} by XBOOLE_0:def 7;
d <> Y by A17;
then d
` is_a_dependent_set_of PA & d` is_a_dependent_set_of PB by A10,A11,Th10;
then A20: d` in XX by A18;
Intersect(XX) c= d`
proof
let X1 be object;
assume X1 in Intersect(XX);
then X1 in meet XX by A4,SETFAM_1:def 9;
hence thesis by A20,SETFAM_1:def 1;
end;
then A21: d /\ Intersect(XX) = {} by A19,XBOOLE_1:3,26;
d /\ d c= d /\ Intersect(XX) by A9,XBOOLE_1:26;
then A22: union Ad = {} by A14,A21;
consider ad being object such that
A23: ad in Ad by A13,XBOOLE_0:def 1;
A24: ad<>{} by A2,A12,A23;
reconsider ad as set by TARSKI:1;
ad c= {} by A22,A23,ZFMISC_1:74;
hence thesis by A24;
end;
end;
hence thesis by A4,A5,A7,A8,SETFAM_1:def 9;
end;
theorem
for P being a_partition of Y for y being Element of Y
ex A being Subset of Y st y in A & A in P
proof
let P be a_partition of Y;
let y be Element of Y;
consider R being Equivalence_Relation of Y such that
A1: P = Class R by EQREL_1:34;
take Class(R,y);
thus y in Class(R,y) by EQREL_1:20;
thus thesis by A1,EQREL_1:def 3;
end;
definition
let Y be set;
func PARTITIONS(Y) -> set means
:Def3:
for x being set holds x in it iff x is a_partition of Y;
existence
proof
defpred P[set] means $1 is a_partition of Y;
consider X being set such that
A1: for x being set holds x in X iff x in bool bool Y & P[x] from
XFAMILY:sch 1;
take X;
let x be set;
thus x in X implies x is a_partition of Y by A1;
assume x is a_partition of Y;
hence thesis by A1;
end;
uniqueness
proof
let A1,A2 be set such that
A2: for x being set holds x in A1 iff x is a_partition of Y and
A3: for x being set holds x in A2 iff x is a_partition of Y;
now
let y be object;
y in A1 iff y is a_partition of Y by A2;
hence y in A1 iff y in A2 by A3;
end;
hence thesis by TARSKI:2;
end;
end;
registration
let Y be set;
cluster PARTITIONS(Y) -> non empty;
coherence
proof
per cases;
suppose
Y={};
hence thesis by Def3,EQREL_1:45;
end;
suppose
Y<>{};
then reconsider Y as non empty set;
A1: {Y} is Subset-Family of Y & union {Y} = Y by ZFMISC_1:25,68;
for A being Subset of Y st A in {Y} holds A<>{} &
for B being Subset of Y st B in {Y} holds A = B or A misses B
proof
let A be Subset of Y;
assume
A2: A in {Y};
then A3: A=Y by TARSKI:def 1;
thus A<>{} by A2,TARSKI:def 1;
let B be Subset of Y;
assume B in {Y};
hence thesis by A3,TARSKI:def 1;
end;
then {Y} is a_partition of Y by A1,EQREL_1:def 4;
hence thesis by Def3;
end;
end;
end;
begin
::Chap. 2 Join and Meet Operation between Partitions
definition
let Y;
let PA,PB be a_partition of Y;
func PA '/\' PB -> a_partition of Y equals
INTERSECTION(PA,PB) \ {{}};
correctness
proof
{} c= Y;
then reconsider e = {{}} as Subset-Family of Y by ZFMISC_1:31;
set a = INTERSECTION(PA,PB);
for z being object st z in a holds z in bool Y
proof
let z be object;
reconsider zz=z as set by TARSKI:1;
assume z in a;
then ex M,N being set st M in PA & N in PB & z = M /\ N by SETFAM_1:def 5;
then zz c= Y by XBOOLE_1:108;
hence thesis;
end;
then
reconsider a = INTERSECTION(PA,PB) as Subset-Family of Y by TARSKI:def 3;
reconsider ia = a \ e as Subset-Family of Y;
A1: union PA = Y & union PB = Y by EQREL_1:def 4;
A2: union ia = union INTERSECTION(PA,PB) by Th2
.= union PA /\ union PB by SETFAM_1:28
.= Y by A1;
for A being Subset of Y st A in ia holds A<>{} &
for B being Subset of Y st B in ia holds A = B or A misses B
proof
let A be Subset of Y;
assume
A3: A in ia;
then A4: not A in {{}} by XBOOLE_0:def 5;
A5: A in INTERSECTION(PA,PB) by A3,XBOOLE_0:def 5;
for B being Subset of Y st B in ia holds A = B or A misses B
proof
A6: for
m,n,o,p being set holds (m /\ n) /\ (o /\ p) = m /\ ( n /\ o /\ p)
proof
let m,n,o,p be set;
(m /\ n) /\ (o /\ p) = m /\ ( n /\ (o /\ p) ) by XBOOLE_1:16
.= m /\ ( n /\ o /\ p) by XBOOLE_1:16;
hence thesis;
end;
let B be Subset of Y;
assume B in ia;
then B in INTERSECTION(PA,PB) by XBOOLE_0:def 5;
then consider M,N being set such that
A7: M in PA & N in PB and
A8: B = M /\ N by SETFAM_1:def 5;
consider S,T being set such that
A9: S in PA & T in PB and
A10: A = S /\ T by A5,SETFAM_1:def 5;
M /\ N = S /\ T or (not M = S or not N = T );
then M /\ N = S /\ T or (M misses S or N misses T)
by A7,A9,EQREL_1:def 4;
then A11: M /\ N = S /\ T or ( M /\ S = {} or N /\ T = {}) by XBOOLE_0:def 7
;
(M /\ S) /\ (N /\ T) = M /\ ( S /\ N /\ T ) by A6
.=( M /\ N ) /\ ( S /\ T ) by A6;
hence thesis by A8,A10,A11,XBOOLE_0:def 7;
end;
hence thesis by A4,TARSKI:def 1;
end;
hence thesis by A2,EQREL_1:def 4;
end;
commutativity;
end;
theorem
for PA being a_partition of Y holds PA '/\' PA = PA
proof
let PA be a_partition of Y;
for u being set st u in INTERSECTION(PA,PA) \ {{}} ex v being set
st v in PA & u c= v
proof
let u be set;
assume u in INTERSECTION(PA,PA) \ {{}};
then consider v,u2 being set such that
A1: v in PA and u2 in PA and
A2: u = v /\ u2 by SETFAM_1:def 5;
take v;
thus thesis by A1,A2,XBOOLE_1:17;
end;
then A3: INTERSECTION(PA,PA) \ {{}} '<' PA by SETFAM_1:def 2;
for u being set st u in PA
ex v being set st v in INTERSECTION(PA,PA) \ {{}} & u c= v
proof
let u be set;
assume
A4: u in PA;
then A5: u <> {} by EQREL_1:def 4;
set v = u /\ u;
A6: not v in {{}} by A5,TARSKI:def 1;
v in INTERSECTION(PA,PA) by A4,SETFAM_1:def 5;
then v in INTERSECTION(PA,PA) \ {{}} by A6,XBOOLE_0:def 5;
hence thesis;
end;
then PA '<' INTERSECTION(PA,PA) \ {{}} by SETFAM_1:def 2;
hence thesis by A3,Th4;
end;
theorem
for PA,PB,PC being a_partition of Y holds
(PA '/\' PB) '/\' PC = PA '/\' (PB '/\' PC)
proof
let PA,PB,PC be a_partition of Y;
consider PD,PE being a_partition of Y such that
A1: PD = PA '/\' PB and
A2: PE = PB '/\' PC;
for u being set st u in PD '/\' PC
ex v being set st v in PA '/\' PE & u c= v
proof
let u be set;
assume
A3: u in PD '/\' PC;
then consider u1, u2 being set such that
A4: u1 in PD and
A5: u2 in PC and
A6: u = u1 /\ u2 by SETFAM_1:def 5;
consider u3, u4 being set such that
A7: u3 in PA and
A8: u4 in PB and
A9: u1 = u3 /\ u4 by A1,A4,SETFAM_1:def 5;
consider v, v1,v2 being set such that
A10: v1 = u4 /\ u2 and
A11: v2 = u3 and
A12: v = u3 /\ u4 /\ u2;
A13: v = v2 /\ v1 by A10,A11,A12,XBOOLE_1:16;
A14: v1 in INTERSECTION(PB,PC) by A5,A8,A10,SETFAM_1:def 5;
A15: not u in {{}} by A3,XBOOLE_0:def 5;
u = u3 /\ v1 by A6,A9,A10,XBOOLE_1:16;
then v1 <> {} by A15,TARSKI:def 1;
then not v1 in {{}} by TARSKI:def 1;
then v1 in INTERSECTION(PB,PC) \ {{}} by A14,XBOOLE_0:def 5;
then A16: v in INTERSECTION(PA,PE) by A2,A7,A11,A13,SETFAM_1:def 5;
take v;
thus thesis by A6,A9,A12,A15,A16,XBOOLE_0:def 5;
end;
then A17: PD '/\' PC '<' PA '/\' PE by SETFAM_1:def 2;
for u being set st u in PA '/\' PE
ex v being set st v in PD '/\' PC & u c= v
proof
let u be set;
assume
A18: u in PA '/\' PE;
then consider u1,u2 being set such that
A19: u1 in PA and
A20: u2 in PE and
A21: u = u1 /\ u2 by SETFAM_1:def 5;
consider u3,u4 being set such that
A22: u3 in PB and
A23: u4 in PC and
A24: u2 = u3 /\ u4 by A2,A20,SETFAM_1:def 5;
A25: u = u1 /\ u3 /\ u4 by A21,A24,XBOOLE_1:16;
consider v, v1,v2 being set such that
A26: v1 = u1 /\ u3 and v2 = u4 and
A27: v = u1 /\ u3 /\ u4;
A28: v1 in INTERSECTION(PA,PB) by A19,A22,A26,SETFAM_1:def 5;
A29: not u in {{}} by A18,XBOOLE_0:def 5;
then v1 <> {} by A25,A26,TARSKI:def 1;
then not v1 in {{}} by TARSKI:def 1;
then v1 in INTERSECTION(PA,PB) \ {{}} by A28,XBOOLE_0:def 5;
then A30: v in INTERSECTION(PD,PC) by A1,A23,A26,A27,SETFAM_1:def 5;
take v;
thus thesis by A25,A27,A29,A30,XBOOLE_0:def 5;
end;
then PA '/\' PE '<' PD '/\' PC by SETFAM_1:def 2;
hence thesis by A1,A2,A17,Th4;
end;
theorem Th15:
for PA,PB being a_partition of Y holds PA '>' PA '/\' PB
proof
let PA,PB be a_partition of Y;
for u being set st u in PA '/\' PB ex v being set st v in PA & u c= v
proof
let u be set;
assume u in PA '/\' PB;
then consider u1,u2 being set such that
A1: u1 in PA and u2 in PB and
A2: u = u1 /\ u2 by SETFAM_1:def 5;
take u1;
thus thesis by A1,A2,XBOOLE_1:17;
end;
hence PA '>' PA '/\' PB by SETFAM_1:def 2;
end;
definition
let Y;
let PA,PB be a_partition of Y;
func PA '\/' PB -> a_partition of Y means
:Def5:
for d holds d in it iff d is_min_depend PA,PB;
existence
proof
A1: union PA = Y by EQREL_1:def 4;
A2: for A being set st A in PA holds A<>{} & for B being set st B in PA
holds A=B or A misses B by EQREL_1:def 4;
defpred P[set] means $1 is_min_depend PA,PB;
consider X being set such that
A3: for d being set holds d in X iff d in bool Y & P[d] from XFAMILY:sch 1;
A4: for d being set holds d in X iff d is_min_depend PA,PB
proof
let d be set;
for d being set holds d is_min_depend PA,PB implies d in bool Y
proof
let d be set;
assume d is_min_depend PA,PB;
then d is_a_dependent_set_of PA;
then ex A being set st A c= PA & A<>{} & d = union A;
then d c= union PA by ZFMISC_1:77;
hence thesis by A1;
end;
then
d is_min_depend PA,PB implies d is_min_depend PA,PB & d in bool Y;
hence thesis by A3;
end;
take X;
A5: Y c= union X
proof
let y be object;
assume y in Y;
then consider a being Subset of Y such that
A6: y in a and
A7: a is_min_depend PA,PB by Th11;
a in X by A4,A7;
hence thesis by A6,TARSKI:def 4;
end;
union X c= Y
proof
let y be object;
assume y in union X;
then consider a being set such that
A8: y in a and
A9: a in X by TARSKI:def 4;
a is_min_depend PA,PB by A4,A9;
then a is_a_dependent_set_of PA;
then ex A being set st A c= PA & A<>{} & a = union A;
then a c= Y by A1,ZFMISC_1:77;
hence thesis by A8;
end;
then A10: union X = Y by A5,XBOOLE_0:def 10;
A11: for A being Subset of Y st A in X holds A<>{} &
for B being Subset of Y st B in X holds A=B or A misses B
proof
let A be Subset of Y;
assume A in X;
then A12: A is_min_depend PA,PB by A4;
then A13: A is_a_dependent_set_of PA;
A14: A is_a_dependent_set_of PB by A12;
consider Aa being set such that
A15: Aa c= PA and
A16: Aa<>{} and
A17: A = union Aa by A13;
consider aa being object such that
A18: aa in Aa by A16,XBOOLE_0:def 1;
A19: aa<>{} by A2,A15,A18;
reconsider aa as set by TARSKI:1;
aa c= union Aa by A18,ZFMISC_1:74;
hence A<>{} by A17,A19;
let B be Subset of Y;
assume B in X;
then A20: B is_min_depend PA,PB by A4;
then
A21: B is_a_dependent_set_of PA & B is_a_dependent_set_of PB;
now
assume A meets B;
then A22: A
/\ B is_a_dependent_set_of PA & A /\ B is_a_dependent_set_of PB by A13,A14,A21
,Th9;
A /\ B c= A by XBOOLE_1:17;
then A23: A /\ B = A by A12,A22;
A24: A c= B
by A23,XBOOLE_0:def 4;
A /\ B c= B by XBOOLE_1:17;
then A25: A /\ B = B by A20,A22;
B c= A
by A25,XBOOLE_0:def 4;
hence A=B by A24,XBOOLE_0:def 10;
end;
hence thesis;
end;
X c= bool Y
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume x in X;
then xx is_min_depend PA,PB by A4;
then xx is_a_dependent_set_of PA;
then ex a being set st a c= PA & a<>{} & x = union a;
then xx c= Y by A1,ZFMISC_1:77;
hence thesis;
end;
then reconsider X as Subset-Family of Y;
X is a_partition of Y by A10,A11,EQREL_1:def 4;
hence thesis by A4;
end;
uniqueness
proof
let A1,A2 be a_partition of Y such that
A26: for x being set holds x in A1 iff x is_min_depend PA,PB and
A27: for x being set holds x in A2 iff x is_min_depend PA,PB;
for y being object holds y in A1 iff y in A2 by A26,A27;
hence thesis by TARSKI:2;
end;
commutativity
proof
let IT,PA,PB be a_partition of Y;
A28: for a be set st a is_min_depend PA,PB holds a is_min_depend PB,PA;
A29: for a be set st a is_min_depend PB,PA holds a is_min_depend PA,PB;
(for d being set holds (d in a) iff (d is_min_depend PA,PB)) implies
for d being set holds d in a iff d is_min_depend PB,PA
by A28,A29;
hence thesis;
end;
end;
theorem Th16:
for PA,PB being a_partition of Y holds PA '<' PA '\/' PB
proof
thus PA '<' PA '\/' PB
proof
for a being set st a in PA ex b being set st b in PA '\/' PB & a c= b
proof
let a be set;
assume
A1: a in PA;
then A2: a<>{} by EQREL_1:def 4;
set x = the Element of a;
A3: x in Y by A1,A2,TARSKI:def 3;
union (PA '\/' PB) = Y by EQREL_1:def 4;
then consider b being set such that
A4: x in b and
A5: b in PA '\/' PB by A3,TARSKI:def 4;
b is_min_depend PA,PB by A5,Def5;
then b is_a_dependent_set_of PA;
then consider B being set such that
A6: B c= PA and B<>{} and
A7: b = union B;
a in B
proof
consider u being set such that
A8: x in u and
A9: u in B by A4,A7,TARSKI:def 4;
a /\ u <> {} by A2,A8,XBOOLE_0:def 4;
then A10: not a misses u by XBOOLE_0:def 7;
u in PA by A6,A9;
hence thesis by A1,A9,A10,EQREL_1:def 4;
end;
hence thesis by A5,A7,ZFMISC_1:74;
end;
hence thesis by SETFAM_1:def 2;
end;
end;
theorem
for PA being a_partition of Y holds PA '\/' PA = PA
proof
let PA be a_partition of Y;
A1: PA '<' PA '\/' PA by Th16;
for a being set st a in PA '\/' PA ex b being set st b in PA & a c= b
proof
let a be set;
assume
A2: a in PA '\/' PA;
then A3: a is_min_depend PA,PA by Def5;
then a is_a_dependent_set_of PA;
then consider B being set such that
A4: B c= PA and B<>{} and
A5: a = union B;
A6: a <> {} by A2,EQREL_1:def 4;
set x = the Element of a;
x in a by A6;
then x in Y by A2;
then x in union PA by EQREL_1:def 4;
then consider b being set such that
A7: x in b and
A8: b in PA by TARSKI:def 4;
b in B
proof
consider u being set such that
A9: x in u and
A10: u in B by A5,A6,TARSKI:def 4;
b /\ u <> {} by A7,A9,XBOOLE_0:def 4;
then A11: not b misses u by XBOOLE_0:def 7;
u in PA by A4,A10;
hence thesis by A8,A10,A11,EQREL_1:def 4;
end;
then A12: b c= a by A5,ZFMISC_1:74;
b is_a_dependent_set_of PA by A8,Th6;
then a = b by A3,A12;
hence thesis by A8;
end;
then PA '\/' PA '<' PA by SETFAM_1:def 2;
hence thesis by A1,Th4;
end;
theorem Th18:
for PA,PC being a_partition of Y st
PA '<' PC & x in PC & z0 in PA & t in x & t in z0 holds z0 c= x
proof
let PA,PC be a_partition of Y;
assume that
A1: PA '<' PC and
A2: x in PC and
A3: z0 in PA and
A4: t in x & t in z0;
consider b such that
A5: b in PC and
A6: z0 c= b by A1,A3,SETFAM_1:def 2;
x = b or x misses b by A2,A5,EQREL_1:def 4;
hence thesis by A4,A6,XBOOLE_0:3;
end;
theorem
for PA,PB being a_partition of Y st
x in (PA '\/' PB) & z0 in PA & t in x & t in z0 holds z0 c= x by Th16,Th18;
begin
::Chap. 3 Partitions and Equivalence Relations
definition
let Y;
let PA be a_partition of Y;
func ERl PA -> Equivalence_Relation of Y means
:Def6:
for x1,x2 being object holds
[x1,x2] in it iff ex A st A in PA & x1 in A & x2 in A;
existence
proof
A1: union PA = Y by EQREL_1:def 4;
ex RA being Equivalence_Relation of Y st
for x,y being object holds [x,y] in RA iff ex A st A in PA & x in A & y in A
proof
defpred P[object,object] means ex A st A in PA & $1 in A & $2 in A;
A2: for x being object st x in Y holds P[x,x]
proof
let x be object;
assume x in Y;
then ex Z being set st x in Z & Z in PA by A1,TARSKI:def 4;
then consider Z such that
A3: x in Z and
A4: Z in PA;
reconsider A = Z as Subset of Y by A4;
take A;
thus thesis by A3,A4;
end;
A5: for x,y being object st P[x,y] holds P[y,x];
A6: for x,y,z being object st P[x,y] & P[y,z] holds P[x,z]
proof
let x,y,z be object;
given A such that
A7: A in PA and
A8: x in A & y in A;
given B such that
A9: B in PA and
A10: y in B & z in B;
A = B or A misses B by A7,A9,EQREL_1:def 4;
hence thesis by A7,A8,A10,XBOOLE_0:3;
end;
consider RA being Equivalence_Relation of Y such that
A11: for x,y being object holds [x,y] in RA iff
x in Y & y in Y & P[x,y] from EQREL_1:sch 1(A2,A5,A6);
take RA;
thus thesis by A11;
end;
hence thesis;
end;
uniqueness
proof
let f1,f2 be Equivalence_Relation of Y such that
A12: for x1,x2 being object holds [x1,x2] in f1 iff
ex A st A in PA & x1 in A & x2 in A and
A13: for x1,x2 being object holds [x1,x2] in f2 iff
ex A st A in PA & x1 in A & x2 in A;
let x1,x2 be object;
[x1,x2] in f1 iff ex A st A in PA & x1 in A & x2 in A by A12;
hence thesis by A13;
end;
end;
definition
let Y;
defpred P[object,object] means ex PA st PA = $1 & $2 = ERl PA;
A1: for x being object st x in PARTITIONS(Y) ex z being object st P[x,z]
proof
let x be object;
assume x in PARTITIONS(Y);
then reconsider x as a_partition of Y by Def3;
take ERl x;
thus thesis;
end;
func Rel(Y) -> Function means
dom it = PARTITIONS(Y) &
for x being object st x in PARTITIONS(Y) ex PA st PA = x & it.x = ERl PA;
existence
proof
thus ex f being Function st dom f = PARTITIONS(Y)
& for x being object st x in PARTITIONS(Y) holds P[x,f.x]
from CLASSES1:sch 1(A1);
end;
uniqueness
proof
let f1,f2 be Function such that
A2: dom f1 = PARTITIONS(Y) and
A3: for x being object st x in PARTITIONS(Y) ex PA st PA = x & f1.x = ERl PA
and
A4: dom f2 = PARTITIONS(Y) and
A5: for x being object st x in PARTITIONS(Y) ex PA st PA = x & f2.x = ERl PA;
for z being object st z in PARTITIONS(Y) holds f1.z = f2.z
proof
let x be object;
assume x in PARTITIONS(Y);
then (
ex PA st PA = x & f1.x = ERl PA)& ex PA st PA = x & f2.x = ERl PA by A3,A5;
hence thesis;
end;
hence thesis by A2,A4,FUNCT_1:2;
end;
end;
theorem Th20:
for PA,PB being a_partition of Y holds PA '<' PB iff ERl(PA) c= ERl(PB)
proof
let PA,PB be a_partition of Y;
set RA = ERl PA, RB = ERl PB;
hereby
assume
A1: PA '<' PB;
for x1,x2 being object holds [x1,x2] in RA implies [x1,x2] in RB
proof
let x1,x2 be object;
assume [x1,x2] in RA;
then consider A being Subset of Y such that
A2: A in PA and
A3: x1 in A & x2 in A by Def6;
ex y st y in PB & A c= y by A1,A2,SETFAM_1:def 2;
hence thesis by A3,Def6;
end;
hence ERl(PA) c= ERl(PB);
end;
assume
A4: ERl(PA) c= ERl(PB);
for x st x in PA ex y st y in PB & x c= y
proof
let x;
assume
A5: x in PA;
then A6: x<>{} by EQREL_1:def 4;
set x0 = the Element of x;
set y={z where z is Element of Y:[x0,z] in ERl(PB)};
A7: x c= y
proof
let x1 be object;
assume
A8: x1 in x;
then [x0,x1] in RA by A5,Def6;
hence thesis by A4,A5,A8;
end;
set x1 = the Element of x;
[x0,x1] in RA by A5,A6,Def6;
then consider B being Subset of Y such that
A9: B in PB and
A10: x0 in B and x1 in B by A4,Def6;
A11: y c= B
proof
let x2 be object;
assume x2 in y;
then ex z being Element of Y st z=x2 & [x0,z] in ERl(PB);
then consider B2 being Subset of Y such that
A12: B2 in PB and
A13: x0 in B2 and
A14: x2 in B2 by Def6;
B2 meets B by A10,A13,XBOOLE_0:3;
hence thesis by A9,A12,A14,EQREL_1:def 4;
end;
B c= y
proof
let x2 be object;
assume
A15: x2 in B;
then [x0,x2] in RB by A9,A10,Def6;
hence thesis by A15;
end;
then y=B by A11,XBOOLE_0:def 10;
hence thesis by A7,A9;
end;
hence thesis by SETFAM_1:def 2;
end;
theorem Th21:
for PA,PB being a_partition of Y,p0,x,y being set,
f being FinSequence of Y st p0 c= Y &
x in p0 & f.1=x & f.len f=y & 1 <= len f & (for i st 1<=i & i{} and
A8: p0 = union A1 by A5;
consider B1 being set such that
A9: B1 c= PB and B1<>{} and
A10: p0 = union B1 by A6;
A11: union PA = Y by EQREL_1:def 4;
A12: for A being set st A in PA holds A<>{} & for B being set st B in PA
holds A=B or A misses B by EQREL_1:def 4;
A13: for A being set st A in PB holds A<>{} & for B being set st B in PB
holds A=B or A misses B by EQREL_1:def 4;
for k st 1 <= k & k <= len f holds f.k in p0
proof
defpred P[Nat] means 1 <= $1 & $1 <= len f implies f.$1 in p0;
A14: P[0];
A15: for k st P[k] holds P[k+1]
proof
let k;
assume
A16: P[k];
assume that
A17: 1<=k+1 and
A18: k+1<=len f;
A19: k < len f by A18,NAT_1:13;
A20: 1 <= k or 1 = k + 1 by A17,NAT_1:8;
now per cases by A20;
suppose
A21: 1 <= k;
then consider p2,p3,u being set such that
A22: p2 in PA and
A23: p3 in PB and
A24: f.k in p2 and
A25: u in p2 and
A26: u in p3 and
A27: f.(k+1) in p3 by A4,A19;
consider A being set such that
A28: f.k in A and
A29: A in PA by A1,A11,A16,A18,A21,NAT_1:13,TARSKI:def 4;
A30: p2=A or p2 misses A by A22,A29,EQREL_1:def 4;
consider a being set such that
A31: f.k in a and
A32: a in A1 by A8,A16,A18,A21,NAT_1:13,TARSKI:def 4;
a=p2 or a misses p2 by A7,A12,A22,A32;
then A33: A c= p0 by A8,A24,A28,A30,A31,A32,XBOOLE_0:3,ZFMISC_1:74;
consider B being set such that
A34: u in B and
A35: B in PB by A23,A26;
A36: p3=B or p3 misses B by A23,A35,EQREL_1:def 4;
consider b being set such that
A37: u in b and
A38: b in B1 by A10,A24,A25,A28,A30,A33,TARSKI:def 4,XBOOLE_0:3;
p3=b or p3 misses b by A9,A13,A23,A38;
then B c= p0 by A10,A26,A34,A36,A37,A38,XBOOLE_0:3,ZFMISC_1:74;
hence thesis by A26,A27,A34,A36,XBOOLE_0:3;
end;
suppose
0 = k;
hence thesis by A2;
end;
end;
hence thesis;
end;
thus P[k] from NAT_1:sch 2(A14,A15);
end;
hence thesis by A3;
end;
theorem Th22:
for R1,R2 being Equivalence_Relation of Y,f being FinSequence of Y,
x,y being set st x in Y &
f.1=x & f.len f=y & 1 <= len f &
(for i st 1<=i & i as FinSequence of Y;
A17: len p = 3 by FINSEQ_1:45;
A18: p.1 = f.i & p.3 = f.(i+1) by FINSEQ_1:45;
for j st 1 <= j & j < len p holds [p.j,p.(j+1)] in (R1 \/ R2)
proof
let j;
assume that
A19: 1 <= j and
A20: j < len p;
j < 2+1 by A20,FINSEQ_1:45;
then j <= 2 by NAT_1:13;
then j = 0 or ... or j = 2 by NAT_1:60;
hence thesis by A15,A18,A19,FINSEQ_1:45;
end;
then [f.i,f.(i+1)] in R1 "\/" R2 by A17,A18,EQREL_1:28;
hence thesis by A7,A9,A13,EQREL_1:7,NAT_1:13;
end;
suppose
A21: 0 = i;
[f.1,f.1] in R1 by A1,A2,EQREL_1:5;
then [f.1,f.1] in R1 \/ R2 by XBOOLE_0:def 3;
hence thesis by A12,A21;
end;
end;
hence thesis;
end;
thus P[i] from NAT_1:sch 2(A5,A6);
end;
hence thesis by A2,A3;
end;
theorem Th23:
for PA,PB being a_partition of Y holds ERl(PA '\/' PB) = ERl(PA) "\/" ERl(PB)
proof
let PA,PB be a_partition of Y;
A1: PA is_finer_than (PA '\/' PB) by Th16;
A2: PB is_finer_than (PA '\/' PB) by Th16;
A3: union PA = Y by EQREL_1:def 4;
A4: union PB = Y by EQREL_1:def 4;
A5: ERl(PA '\/' PB) c= (ERl(PA) "\/" ERl(PB))
proof
let x,y be object;
assume [x,y] in ERl(PA '\/' PB);
then consider p0 being Subset of Y such that
A6: p0 in (PA '\/' PB) and
A7: x in p0 and
A8: y in p0 by Def6;
A9: p0 is_min_depend PA,PB by A6,Def5;
then A10: p0 is_a_dependent_set_of PA;
A11: p0 is_a_dependent_set_of PB by A9;
consider A1 being set such that
A12: A1 c= PA and A1<>{} and
A13: p0 = union A1 by A10;
consider a being set such that
A14: x in a and
A15: a in A1 by A7,A13,TARSKI:def 4;
reconsider Ca={ p where p is Element of PA: ex f being FinSequence of Y st
1<=len f & f.1=x & f.len f in p & for i st 1<=i & i {} } as set;
reconsider x9 = x as Element of Y by A7;
reconsider fx=<*x9*> as FinSequence of Y;
A16: fx.1=x by FINSEQ_1:def 8;
then A17: fx.len fx in a by A14,FINSEQ_1:40;
1
<=len fx & for i st 1<=i & i {} by A19,A21,XBOOLE_0:def 4;
then A23: z5 in Cb by A20,A22;
Ca c= PA
proof
let z be object;
assume z in Ca;
then ex p being Element of PA st z=p & ex f being FinSequence of Y st
1<=len f & f.1=x & f.len f in p & for i st 1<=i & i {} by A26,A28,XBOOLE_0:def 4;
then z in Cb by A27,A29;
hence thesis by A28,TARSKI:def 4;
end;
union Cb c= pb
proof
let x1 be object;
assume x1 in union Cb;
then consider y1 being set such that
A30: x1 in y1 and
A31: y1 in Cb by TARSKI:def 4;
A32: ex p being Element of PB st y1=p & ex q being set st q
in Ca & p /\ q <> {} by A31;
then consider q being set such that
A33: q in Ca and
A34: y1 /\ q <> {};
consider pd being set such that
A35: x1 in pd and
A36: pd in PA by A3,A30,A32,TARSKI:def 4;
A37: ex pp being Element of PA st q=pp & ex f being
FinSequence of Y st 1<=len f & f.1=x & f.len f in pp & for i st 1<=i & i as FinSequence of Y;
len f = len fd + len <*x1*> by FINSEQ_1:22;
then A42: len f = len fd + 1 by FINSEQ_1:40;
1+1<=len fd + 1 by A38,XREAL_1:6;
then A43: 1<=len f by A42,XXREAL_0:2;
A44: f.(len fd + 1) in y1 by A30,FINSEQ_1:42;
y1 meets q by A34,XBOOLE_0:def 7;
then consider z0 being object such that
A45: z0 in y1 & z0 in q by XBOOLE_0:3;
A46: z0 in (y1 /\ q) by A45,XBOOLE_0:def 4;
A47: dom fd = Seg len fd by FINSEQ_1:def 3;
A48: for k being Nat st 1 <= k & k <= len fd holds f.k=fd.k
proof
let k be Nat;
assume 1 <= k & k <= len fd;
then k in dom fd by A47,FINSEQ_1:1;
hence thesis by FINSEQ_1:def 7;
end;
then
A49: (fd^<*x1*>).(len fd + 1)=x1 & f.1=x by A38,A39,FINSEQ_1:42;
A50: f.len fd in q by A38,A40,A48;
A51: for i st 1<=i & i {};
hence thesis;
end;
then A58: pb is_a_dependent_set_of PB by A23,A57;
now
assume not pb c= p0;
then pb \ p0 <> {} by XBOOLE_1:37;
then consider x1 being object such that
A59: x1 in (pb \ p0) by XBOOLE_0:def 1;
A60: not x1 in p0 by A59,XBOOLE_0:def 5;
consider y1 being set such that
A61: x1 in y1 and
A62: y1 in Cb by A57,A59,TARSKI:def 4;
A63: ex p being Element of PB st y1=p & ex q being set st q
in Ca & p /\ q <> {} by A62;
then consider q being set such that
A64: q in Ca and
A65: y1 /\ q <> {};
A66: ex pp being Element of PA st q=pp & ex f being
FinSequence of Y st 1<=len f & f.1=x & f.len f in pp & for i st 1<=i & i as FinSequence of Y;
len f = len fd + len <*x1*> by FINSEQ_1:22;
then A71: len f = len fd + 1 by FINSEQ_1:40;
1+1<=len fd + 1 by A67,XREAL_1:6;
then A72: 1<=len f by A71,XXREAL_0:2;
A73: f.(len fd + 1) in y1 by A61,FINSEQ_1:42;
y1 meets q by A65,XBOOLE_0:def 7;
then consider z0 being object such that
A74: z0 in y1 & z0 in q by XBOOLE_0:3;
A75: z0 in (y1 /\ q) by A74,XBOOLE_0:def 4;
A76: dom fd = Seg len fd by FINSEQ_1:def 3;
A77: for k being Nat st 1 <= k & k <= len fd holds f.k=fd.k
proof
let k be Nat;
assume 1 <= k & k <= len fd;
then k in dom fd by A76,FINSEQ_1:1;
hence thesis by FINSEQ_1:def 7;
end;
then
A78: (fd^<*x1*>).(len fd + 1)=x1 & f.1=x by A67,A68,FINSEQ_1:42;
A79: f.len fd in q by A67,A69,A77;
A80: for i st 1<=i & i {} by A91;
then consider q being set such that
A93: q in Ca and
A94: y1 /\ q <> {};
A95: ex pp being Element of PA st q=pp & ex f being
FinSequence of Y st 1<=len f & f.1=x & f.len f in pp & for i st 1<=i & i as FinSequence of Y;
len f = len fd + len <*y9*> by FINSEQ_1:22;
then A100: len f = len fd + 1 by FINSEQ_1:40;
then A101: 1+1<=len f by A96,XREAL_1:6;
A102: f.(len fd + 1) in y1 by A90,FINSEQ_1:42;
y1 meets q by A94,XBOOLE_0:def 7;
then consider z0 being object such that
A103: z0 in y1 & z0 in q by XBOOLE_0:3;
A104: z0 in (y1 /\ q) by A103,XBOOLE_0:def 4;
A105: dom fd = Seg len fd by FINSEQ_1:def 3;
A106: for k being Nat st 1 <= k & k <= len fd holds f.k=fd.k
proof
let k be Nat;
assume 1 <= k & k <= len fd;
then k in dom fd by A105,FINSEQ_1:1;
hence thesis by FINSEQ_1:def 7;
end;
then A107: f.len fd in q by A96,A98;
A108: for i st 1<=i & i).(len fd + 1)=y & 1<=len f by A101,FINSEQ_1:42,XXREAL_0:2;
A116: f.1=x by A96,A97,A106;
for i st 1<=i & i' PA '/\' PB by Th15;
A2: PB '>' PA '/\' PB by Th15;
for x1,x2 being object holds
[x1,x2] in ERl(PA '/\' PB) iff [x1,x2] in (ERl(PA) /\ ERl(PB))
proof
let x1,x2 be object;
hereby
assume [x1,x2] in ERl(PA '/\' PB);
then consider C being Subset of Y such that
A3: C in (PA '/\' PB) and
A4: x1 in C & x2 in C by Def6;
A5: ex A being set st A in PA & C c= A by A1,A3,SETFAM_1:def 2;
A6: ex B being set st B in PB & C c= B by A2,A3,SETFAM_1:def 2;
A7: [x1,x2] in ERl(PA) by A4,A5,Def6;
[x1,x2] in ERl(PB) by A4,A6,Def6;
hence [x1,x2] in (ERl(PA) /\ ERl(PB)) by A7,XBOOLE_0:def 4;
end;
assume
A8: [x1,x2] in (ERl(PA) /\ ERl(PB));
then A9: [x1,x2] in ERl(PA) by XBOOLE_0:def 4;
A10: [x1,x2] in ERl(PB) by A8,XBOOLE_0:def 4;
consider A being Subset of Y such that
A11: A in PA and
A12: x1 in A and
A13: x2 in A by A9,Def6;
consider B being Subset of Y such that
A14: B in PB and
A15: x1 in B and
A16: x2 in B by A10,Def6;
A17: A /\ B <> {} by A12,A15,XBOOLE_0:def 4;
consider C being Subset of Y such that
A18: C = A /\ B;
A19: C in INTERSECTION(PA,PB) by A11,A14,A18,SETFAM_1:def 5;
not C in {{}} by A17,A18,TARSKI:def 1;
then A20: C in INTERSECTION(PA,PB) \ {{}} by A19,XBOOLE_0:def 5;
x1 in C & x2 in C by A12,A13,A15,A16,A18,XBOOLE_0:def 4;
hence thesis by A20,Def6;
end;
hence thesis;
end;
theorem Th25:
for PA,PB being a_partition of Y st ERl(PA) = ERl(PB) holds PA = PB
proof
let PA,PB be a_partition of Y;
assume ERl(PA)=ERl(PB);
then PA '<' PB & PB '<' PA by Th20;
hence thesis by Th4;
end;
theorem
for PA,PB,PC being a_partition of Y holds
(PA '\/' PB) '\/' PC = PA '\/' (PB '\/' PC)
proof
let PA,PB,PC be a_partition of Y;
ERl((PA '\/' PB) '\/' PC) = ERl(PA '\/' PB) "\/" ERl(PC) by Th23
.= (ERl(PA) "\/" ERl(PB)) "\/" ERl(PC) by Th23
.= ERl(PA) "\/" (ERl(PB) "\/" ERl(PC)) by EQREL_1:13
.= ERl(PA) "\/" ERl(PB '\/' PC) by Th23
.= ERl(PA '\/' (PB '\/' PC)) by Th23;
hence thesis by Th25;
end;
theorem
for PA,PB being a_partition of Y holds PA '/\' (PA '\/' PB) = PA
proof
let PA,PB be a_partition of Y;
ERl
(PA '/\' (PA '\/' PB)) = ERl(PA) /\ ERl(PA '\/' PB) & ERl(PA) /\ ERl(PA
'\/' PB) = ERl(PA) /\ (ERl(PA) "\/" ERl(PB)) by Th23,Th24;
hence thesis by Th25,EQREL_1:16;
end;
theorem
for PA,PB being a_partition of Y holds PA '\/' (PA '/\' PB) = PA
proof
let PA,PB be a_partition of Y;
ERl
(PA '\/' (PA '/\' PB)) = ERl(PA) "\/" ERl(PA '/\' PB) & ERl(PA) "\/" ERl(
PA '/\' PB) = ERl(PA) "\/" (ERl(PA) /\ ERl(PB)) by Th23,Th24;
hence thesis by Th25,EQREL_1:17;
end;
theorem Th29:
for PA,PB,PC being a_partition of Y st
PA '<' PC & PB '<' PC holds PA '\/' PB '<' PC
proof
let PA,PB,PC be a_partition of Y;
assume PA '<' PC & PB '<' PC;
then A1: ERl(PA) c= ERl(PC) & ERl(PB) c= ERl(PC) by Th20;
A2: ERl(PA '\/' PB) = ERl(PA) "\/" ERl(PB) by Th23;
ERl(PA) \/ ERl(PB) c= ERl(PC) by A1,XBOOLE_1:8;
then ERl(PA) "\/" ERl(PB) c= ERl(PC) by EQREL_1:def 2;
hence thesis by A2,Th20;
end;
theorem
for PA,PB,PC being a_partition of Y st
PA '>' PC & PB '>' PC holds PA '/\' PB '>' PC
proof
let PA,PB,PC be a_partition of Y;
assume PA '>' PC & PB '>' PC;
then A1: ERl(PC) c= ERl(PA) & ERl(PC) c= ERl(PB) by Th20;
ERl(PA '/\' PB) = ERl(PA) /\ ERl(PB) by Th24;
then ERl(PC) c= ERl(PA '/\' PB) by A1,XBOOLE_1:19;
hence thesis by Th20;
end;
notation
let Y;
synonym %I(Y) for SmallestPartition Y;
end;
definition let Y; ::: technical type update
redefine func %I Y -> a_partition of Y;
correctness;
end;
definition
let Y;
func %O(Y) -> a_partition of Y equals
{Y};
correctness
proof
A1: {Y} is Subset-Family of Y & union {Y} = Y by ZFMISC_1:25,68;
for A st A in {Y} holds A<>{} & for B st B in {Y} holds A = B or A misses B
proof
let A;
assume
A2: A in {Y};
then A3: A=Y by TARSKI:def 1;
thus A<>{} by A2,TARSKI:def 1;
let B;
assume B in {Y};
hence thesis by A3,TARSKI:def 1;
end;
hence thesis by A1,EQREL_1:def 4;
end;
end;
theorem
%I(Y)={B:ex x being set st B={x} & x in Y}
proof
set B0={B:ex x being set st B={x} & x in Y};
A1: %I(Y) c= B0
proof
let a be object;
assume a in %I(Y);
then
a in the set of all {x} where x is Element of Y by EQREL_1:37;
then consider x be Element of Y such that
A2: a={x};
reconsider y=x as Element of Y;
reconsider B={y} as Subset of Y by ZFMISC_1:31;
a=B by A2;
hence thesis;
end;
B0 c= %I(Y)
proof
let x1 be object;
assume x1 in B0;
then ex B st x1=B & ex x being set st B={x} & x in Y;
then x1 in the set of all {z} where z is Element of Y;
hence thesis by EQREL_1:37;
end;
hence thesis by A1,XBOOLE_0:def 10;
end;
theorem Th32:
for PA being a_partition of Y holds %O(Y) '>' PA & PA '>' %I(Y)
proof
let PA be a_partition of Y;
A1: for a being set st a in PA ex b being set st b in %O(Y) & a c= b
proof
let a be set;
assume
A2: a in PA;
then A3: a c= Y;
A4: a<>{} by A2,EQREL_1:def 4;
set x = the Element of a;
A5: x in Y by A2,A4,TARSKI:def 3;
union %O(Y) = Y by EQREL_1:def 4;
then consider b being set such that
x in b and
A6: b in %O(Y) by A5,TARSKI:def 4;
a c= b by A3,A6,TARSKI:def 1;
hence thesis by A6;
end;
for a being set st a in %I(Y) ex b being set st b in PA & a c= b
proof
let a be set;
assume
A7: a in %I(Y);
then a in the set of all {x} where x is Element of Y by EQREL_1:37;
then consider x be Element of Y such that
A8: a={x};
A9: a<>{} by A7,EQREL_1:def 4;
set u = the Element of a;
A10: u=x by A8,TARSKI:def 1;
A11: u in Y by A7,A9,TARSKI:def 3;
union PA = Y by EQREL_1:def 4;
then consider b being set such that
A12: u in b and
A13: b in PA by A11,TARSKI:def 4;
a c= b
by A8,A10,A12,TARSKI:def 1;
hence thesis by A13;
end;
hence thesis by A1,SETFAM_1:def 2;
end;
theorem Th33:
ERl(%O(Y)) = nabla Y
proof
nabla Y c= ERl(%O(Y))
proof
let x1,x2 be object;
assume
A1: [x1,x2] in nabla Y;
A2: Y in %O(Y) by TARSKI:def 1;
x1 in Y & x2 in Y by A1,ZFMISC_1:87;
hence thesis by A2,Def6;
end;
hence thesis;
end;
theorem Th34:
ERl(%I(Y)) = id Y
proof
A1: union %I(Y)=Y by EQREL_1:def 4;
A2: ERl(%I(Y)) c= id Y
proof
let x1,x2 be object;
assume [x1,x2] in ERl(%I(Y));
then consider a being Subset of Y such that
A3: a in %I(Y) and
A4: x1 in a & x2 in a by Def6;
%I(Y) = the set of all {x} where x is Element of Y by EQREL_1:37;
then consider x be Element of Y such that
A5: a={x} by A3;
x1=x & x2=x by A4,A5,TARSKI:def 1;
hence thesis by RELAT_1:def 10;
end;
id Y c= ERl(%I(Y))
proof
let x1,x2 be object;
assume
A6: [x1,x2] in id Y;
then A7: x1 in Y by RELAT_1:def 10;
A8: x1=x2 by A6,RELAT_1:def 10;
ex y being set st x1 in y & y in %I(Y) by A1,A7,TARSKI:def 4;
hence thesis by A8,Def6;
end;
hence thesis by A2;
end;
theorem
%I(Y) '<' %O(Y)
proof
ERl(%O(Y)) = nabla Y by Th33
.= [:Y,Y:];
then ERl(%I(Y)) c= ERl(%O(Y));
hence thesis by Th20;
end;
theorem
for PA being a_partition of Y holds
%O(Y) '\/' PA = %O(Y) & %O(Y) '/\' PA = PA
proof
let PA be a_partition of Y;
A1: ERl(%O(Y) '\/' PA) = ERl(%O(Y)) "\/" ERl(PA) by Th23;
ERl(%O(Y)) = nabla Y by Th33;
then ERl(%O(Y)) \/ ERl(PA) = ERl(%O(Y)) by EQREL_1:1;
then ERl(%O(Y)) c= ERl(%O(Y)) "\/" ERl(PA) by EQREL_1:def 2;
then A2: %O(Y) '<' %O(Y) '\/' PA by A1,Th20;
%O(Y) '>' PA '\/' %O(Y) by Th32;
hence %O(Y) '\/' PA = %O(Y) by A2,Th4;
ERl(%O(Y) '/\' PA) = ERl(%O(Y)) /\ ERl(PA) & ERl(%O(Y)) = nabla Y by Th24
,Th33;
hence %O(Y) '/\' PA = PA by Th25,XBOOLE_1:28;
end;
theorem
for PA being a_partition of Y holds
%I(Y) '\/' PA = PA & %I(Y) '/\' PA = %I(Y)
proof
let PA be a_partition of Y;
A1: ERl(%I(Y)) = id Y by Th34;
A2: ERl
(%I(Y) '\/' PA) = ERl(%I(Y)) "\/" ERl(PA) & ERl(%I(Y)) \/ ERl(PA) c= ERl(
%I(Y)) "\/" ERl(PA) by Th23,EQREL_1:def 2;
A3: ERl(%I(Y)) \/ ERl(PA) = id Y \/ ERl(PA) by Th34;
%I(Y) '<' PA by Th32;
then A4: ERl(%I(Y)) c= ERl(PA) by Th20;
for z being object st z in id Y \/ ERl(PA) holds z in ERl PA
proof
let z be object;
assume
A5: z in id Y \/ ERl(PA);
now per cases by A5,XBOOLE_0:def 3;
case
z in id Y;
hence thesis by A1,A4;
end;
case
z in ERl(PA);
hence thesis;
end;
end;
hence thesis;
end;
then A6: id Y \/ ERl(PA) c= ERl(PA);
ERl(PA) c= id Y \/ ERl(PA) by XBOOLE_1:7;
then id Y \/ ERl(PA) = ERl(PA) by A6,XBOOLE_0:def 10;
then A7: PA '<' %I(Y) '\/' PA by A2,A3,Th20;
%I(Y) '<' PA by Th32;
then %I(Y) '\/' PA '<' PA by Th29;
hence %I(Y) '\/' PA = PA by A7,Th4;
ERl(%I(Y) '/\' PA) = ERl(%I(Y)) /\ ERl(PA) by Th24
.= id Y /\ ERl(PA) by Th34
.= id Y by EQREL_1:10
.= ERl(%I(Y)) by Th34;
hence %I(Y) '/\' PA = %I(Y) by Th25;
end;