Journal of Formalized Mathematics
Volume 13, 2001
University of Bialystok
Copyright (c) 2001 Association of Mizar Users

### Hierarchies and Classifications of Sets

by
Mariusz Giero

MML identifier: TAXONOM2
[ Mizar article, MML identifier index ]

```environ

vocabulary TAXONOM2, TAXONOM1, ABIAN, EQREL_1, TARSKI, PUA2MSS1, REALSET1,
SETFAM_1, YELLOW_1, WELLORD2, HAHNBAN, ZFMISC_1, BOOLE, ORDERS_1,
RELAT_1, RELAT_2, PARTIT1, ORDERS_2, LATTICES, TREES_1;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, SETFAM_1, REALSET1, RELAT_1,
ORDINAL1, RELSET_1, STRUCT_0, ORDERS_1, EQREL_1, ABIAN, PARTIT1,
ORDERS_2, YELLOW_1, TAXONOM1;
constructors ORDERS_2, PUA2MSS1, TAXONOM1, ABIAN, YELLOW_1;
clusters SUBSET_1, RELSET_1, PARTIT1, YELLOW_1, XBOOLE_0, PARTFUN1;
requirements BOOLE, SUBSET;

begin

reserve A for RelStr;
reserve X for non empty set;
reserve PX,PY,PZ,Y,a,b,c,x,y for set;
reserve S1,S2 for Subset of Y;

definition let A;
attr A is with_superior means
:: TAXONOM2:def 1
ex x be Element of A st x is_superior_of the InternalRel of A;
end;

definition let A;
attr A is with_comparable_down means
:: TAXONOM2:def 2
for x,y be Element of A holds
(ex z be Element of A st z <= x & z <= y) implies
(x <= y or y <= x);
end;

theorem :: TAXONOM2:1
for a being set holds
InclPoset {{a}} is non empty reflexive transitive antisymmetric
with_superior with_comparable_down;

definition
cluster non empty reflexive transitive antisymmetric with_superior
with_comparable_down strict RelStr;
end;

definition
mode Tree is with_superior with_comparable_down Poset;
end;

theorem :: TAXONOM2:2
for EqR being Equivalence_Relation of X,
x,y,z be set holds
z in Class(EqR,x) & z in Class(EqR,y) implies Class(EqR,x) = Class(EqR,y);

theorem :: TAXONOM2:3
for P being a_partition of X,
x,y,z being set st x in P & y in P & z in x & z in y holds x = y;

theorem :: TAXONOM2:4
for C,x be set st C is Classification of X & x in union C
holds x c= X;

theorem :: TAXONOM2:5
for C being set st
C is Strong_Classification of X
holds InclPoset union C is Tree;

begin

definition let Y;
attr Y is hierarchic means
:: TAXONOM2:def 3
for x,y be set st (x in Y & y in Y) holds (x c= y or y c= x or x misses y);
end;

definition
cluster trivial -> hierarchic set;
end;

definition
cluster non trivial hierarchic set;
end;

theorem :: TAXONOM2:6
{} is hierarchic;

theorem :: TAXONOM2:7
{{}} is hierarchic;

definition let Y;
mode Hierarchy of Y -> Subset-Family of Y means
:: TAXONOM2:def 4
it is hierarchic;
end;

definition let Y;
attr Y is mutually-disjoint means
:: TAXONOM2:def 5
for x,y be set st x in Y & y in Y & x <> y holds x misses y;
end;

definition let Y;
cluster mutually-disjoint Subset-Family of Y;
end;

theorem :: TAXONOM2:8
{} is mutually-disjoint;

theorem :: TAXONOM2:9
{{}} is mutually-disjoint;

theorem :: TAXONOM2:10
{a} is mutually-disjoint;

definition let Y; let F be Subset-Family of Y;
attr F is T_3 means
:: TAXONOM2:def 6
for A be Subset of Y st A in F
for x be Element of Y st not x in A
ex B be Subset of Y st x in B & B in F & A misses B;
end;

theorem :: TAXONOM2:11
for F be Subset-Family of Y st F = {} holds F is T_3;

definition let Y;
cluster covering T_3 Hierarchy of Y;
end;

definition let Y; let F be Subset-Family of Y;
attr F is lower-bounded means
:: TAXONOM2:def 7

for B being set st
B <> {} & B c= F & B is c=-linear ex c st (c in F & c c= meet B);
end;

theorem :: TAXONOM2:12
for B being mutually-disjoint Subset-Family of Y
st for b be set st b in B holds S1 misses b & Y <> {}
holds B \/ {S1} is mutually-disjoint Subset-Family of Y
& (S1 <> {} implies union (B \/ {S1}) <> union B);

definition let Y; let F be Subset-Family of Y;
attr F is with_max's means
:: TAXONOM2:def 8

for S being Subset of Y st S in F
ex T being Subset of Y st S c= T & T in F &
for V being Subset of Y st T c= V & V in F holds V = Y;
end;

begin

theorem :: TAXONOM2:13
for H being covering Hierarchy of Y st H is with_max's
ex P being a_partition of Y st P c= H;

theorem :: TAXONOM2:14
for H being covering Hierarchy of Y
for B being mutually-disjoint Subset-Family of Y st
B c= H &
for C  being mutually-disjoint Subset-Family of Y st
C c= H & union B c= union C holds B = C
holds B is a_partition of Y;

theorem :: TAXONOM2:15
for H being covering T_3 Hierarchy of Y st H is lower-bounded & not {} in H
for A being Subset of Y
for B being mutually-disjoint Subset-Family of Y st
A in B & B c= H &
for C  being mutually-disjoint Subset-Family of Y st
A in C & C c= H & union B c= union C holds union B = union C
holds B is a_partition of Y;

theorem :: TAXONOM2:16
for H being covering T_3 Hierarchy of Y st H is lower-bounded & not {} in H
for A being Subset of Y
for B being mutually-disjoint Subset-Family of Y st
A in B & B c= H &
for C  being mutually-disjoint Subset-Family of Y st
A in C & C c= H & B c= C holds B = C
holds B is a_partition of Y;

theorem :: TAXONOM2:17
for H being covering T_3 Hierarchy of Y st H is lower-bounded & not {} in H
for A being Subset of Y st A in H
ex P being a_partition of Y st A in P & P c= H;

theorem :: TAXONOM2:18
for h being non empty set
for Pmin being a_partition of X
for hw being set st hw in Pmin & h c= hw
for PS being a_partition of X st h in PS &
for x st x in PS holds (x c= hw or hw c= x or hw misses x)
for PT be set st (for a holds a in PT iff a in PS & a c= hw)
holds PT \/ (Pmin \ {hw}) is a_partition of X &
PT \/ (Pmin \ {hw}) is_finer_than Pmin;

theorem :: TAXONOM2:19
for h being non empty set st h c= X
for Pmax being a_partition of X st
( (ex hy be set st hy in Pmax & hy c= h) &
for x st x in Pmax holds (x c= h or h c= x or h misses x))
for Pb be set st
(for x holds x in Pb iff (x in Pmax & x misses h))
holds Pb \/ {h} is a_partition of X &
Pmax is_finer_than (Pb \/ {h})&
(for Pmin being a_partition of X st Pmax is_finer_than Pmin
for hw being set st hw in Pmin & h c= hw holds
(Pb \/ {h}) is_finer_than Pmin);

theorem :: TAXONOM2:20
for H being covering T_3 Hierarchy of X st H is lower-bounded & not {} in H &
(for C1 be set st
(C1 <> {} & C1 c= PARTITIONS(X) &
(for P1,P2 be set st P1 in C1 & P2 in C1 holds
P1 is_finer_than P2 or P2 is_finer_than P1)) holds
(ex PX,PY st (PX in C1 & PY in C1
& for PZ st PZ in C1 holds
PZ is_finer_than PY & PX is_finer_than PZ)))
ex C being Classification of X st union C = H;

```