:: TAXONOM2 semantic presentation

definition
let A be RelStr ;
attr a1 is with_superior means :: TAXONOM2:def 1
ex x being Element of A st x is_superior_of the InternalRel of A;
end;

:: deftheorem Def1 defines with_superior TAXONOM2:def 1 :
for A being RelStr holds
( A is with_superior iff ex x being Element of A st x is_superior_of the InternalRel of A );

definition
let A be RelStr ;
attr a1 is with_comparable_down means :Def2: :: TAXONOM2:def 2
for x, y being Element of A holds
( for z being Element of A holds
( not z <= x or not z <= y ) or x <= y or y <= x );
end;

:: deftheorem Def2 defines with_comparable_down TAXONOM2:def 2 :
for A being RelStr holds
( A is with_comparable_down iff for x, y being Element of A holds
( for z being Element of A holds
( not z <= x or not z <= y ) or x <= y or y <= x ) );

theorem Th1: :: TAXONOM2:1
for a being set holds
( not InclPoset {{a}} is empty & InclPoset {{a}} is reflexive & InclPoset {{a}} is transitive & InclPoset {{a}} is antisymmetric & InclPoset {{a}} is with_superior & InclPoset {{a}} is with_comparable_down )
proof end;

registration
cluster non empty strict reflexive transitive antisymmetric with_superior with_comparable_down RelStr ;
existence
ex b1 being RelStr st
( not b1 is empty & b1 is reflexive & b1 is transitive & b1 is antisymmetric & b1 is with_superior & b1 is with_comparable_down & b1 is strict )
proof end;
end;

definition
mode Tree is with_superior with_comparable_down Poset;
end;

theorem Th2: :: TAXONOM2:2
for X being non empty set
for EqR being Equivalence_Relation of X
for x, y, z being set st z in Class EqR,x & z in Class EqR,y holds
Class EqR,x = Class EqR,y
proof end;

theorem Th3: :: TAXONOM2:3
for X being non empty set
for P being a_partition of X
for x, y, z being set st x in P & y in P & z in x & z in y holds
x = y
proof end;

theorem Th4: :: TAXONOM2:4
for X being non empty set
for C, x being set st C is Classification of X & x in union C holds
x c= X
proof end;

theorem Th5: :: TAXONOM2:5
for X being non empty set
for C being set st C is Strong_Classification of X holds
InclPoset (union C) is Tree
proof end;

definition
let Y be set ;
attr a1 is hierarchic means :Def3: :: TAXONOM2:def 3
for x, y being set st x in Y & y in Y & not x c= y & not y c= x holds
x misses y;
end;

:: deftheorem Def3 defines hierarchic TAXONOM2:def 3 :
for Y being set holds
( Y is hierarchic iff for x, y being set st x in Y & y in Y & not x c= y & not y c= x holds
x misses y );

registration
cluster trivial -> hierarchic set ;
coherence
for b1 being set st b1 is trivial holds
b1 is hierarchic
proof end;
end;

registration
cluster non trivial hierarchic set ;
existence
ex b1 being set st
( not b1 is trivial & b1 is hierarchic )
proof end;
end;

theorem Th6: :: TAXONOM2:6
{} is hierarchic
proof end;

theorem Th7: :: TAXONOM2:7
{{} } is hierarchic
proof end;

definition
let Y be set ;
mode Hierarchy of c1 -> Subset-Family of a1 means :Def4: :: TAXONOM2:def 4
it is hierarchic;
existence
ex b1 being Subset-Family of Y st b1 is hierarchic
proof end;
end;

:: deftheorem Def4 defines Hierarchy TAXONOM2:def 4 :
for Y being set
for b2 being Subset-Family of Y holds
( b2 is Hierarchy of Y iff b2 is hierarchic );

definition
let Y be set ;
attr a1 is mutually-disjoint means :Def5: :: TAXONOM2:def 5
for x, y being set st x in Y & y in Y & x <> y holds
x misses y;
end;

:: deftheorem Def5 defines mutually-disjoint TAXONOM2:def 5 :
for Y being set holds
( Y is mutually-disjoint iff for x, y being set st x in Y & y in Y & x <> y holds
x misses y );

E84: now
let x be set , y be set ;
assume E29: ( x in {{} } & y in {{} } & x <> y ) ;
( x = {} & y = {} ) by , TARSKI:def 1;
hence x misses y by ;
end;

registration
let Y be set ;
cluster mutually-disjoint Element of bool (bool a1);
existence
ex b1 being Subset-Family of Y st b1 is mutually-disjoint
proof end;
end;

theorem Th8: :: TAXONOM2:8
{} is mutually-disjoint
proof end;

theorem Th9: :: TAXONOM2:9
{{} } is mutually-disjoint by , ;

theorem Th10: :: TAXONOM2:10
for a being set holds {a} is mutually-disjoint
proof end;

E88: now
let Y be set ;
let H be Hierarchy of Y;
assume E29: H is covering ;
let B be mutually-disjoint Subset-Family of Y;
assume E31: ( B c= H & ( for C being mutually-disjoint Subset-Family of Y st C c= H & union B c= union C holds
B = C ) ) ;
thus union B = Y
proof
thus union B c= Y ; :: according to XBOOLE_0:def 10
thus Y c= union B
proof
let y be set ; :: according to TARSKI:def 3
assume E34: y in Y ;
y in union H by , , ABIAN:4;
then consider x being set such that
E35: ( y in x & x in H ) by TARSKI:def 4;
now
assume E47: not y in union B ;
defpred S1[ set ] means a1 c= x;
consider D being set such that
E48: for a being set holds
( a in D iff ( a in B & S1[a] ) ) from XBOOLE_0:sch 1();
set C = (B \ D) \/ {x};
x in {x} by TARSKI:def 1;
then E49: x in (B \ D) \/ {x} by XBOOLE_0:def 2;
E51: B \ D c= (B \ D) \/ {x} by XBOOLE_1:7;
B \ D c= B by XBOOLE_1:36;
then E53: B \ D c= H by , XBOOLE_1:1;
{x} c= H
proof
let s be set ; :: according to TARSKI:def 3
assume E55: s in {x} ;
thus s in H by , , TARSKI:def 1;
end;
then E56: (B \ D) \/ {x} c= H by , XBOOLE_1:8;
then E57: (B \ D) \/ {x} c= bool Y by XBOOLE_1:1;
E58: for p being set st p in B & not p in D & p <> x holds
p misses x
proof
let p be set ;
assume E61: ( p in B & not p in D & p <> x ) ;
E64: not p c= x by , ;
E65: not x c= p by , , , TARSKI:def 4;
H is hierarchic by ;
hence p misses x by , , , Th1, , ;
end;
for p, q being set st p in (B \ D) \/ {x} & q in (B \ D) \/ {x} & p <> q holds
p misses q
proof
let p be set , q be set ;
assume E66: ( p in (B \ D) \/ {x} & q in (B \ D) \/ {x} & p <> q ) ;
per cases ( p in B \ D or p in {x} ) by , XBOOLE_0:def 2;
suppose p in B \ D ;
then E67: ( p in B & not p in D ) by XBOOLE_0:def 4;
per cases ( q in B \ D or q in {x} ) by , XBOOLE_0:def 2;
suppose q in B \ D ;
then ( q in B & not q in D ) by XBOOLE_0:def 4;
hence p misses q by , Th1, ;
end;
suppose q in {x} ;
then q = x by TARSKI:def 1;
hence p misses q by Def2, , Th1;
end;
end;
end;
suppose p in {x} ;
then E68: p = x by TARSKI:def 1;
per cases ( q in B \ D or q in {x} ) by , XBOOLE_0:def 2;
suppose q in B \ D ;
then ( q in B & not q in D ) by XBOOLE_0:def 4;
hence p misses q by Def2, , Th1;
end;
suppose q in {x} ;
hence p misses q by , Th1, TARSKI:def 1;
end;
end;
end;
end;
end;
then E70: (B \ D) \/ {x} is mutually-disjoint Subset-Family of Y by , ;
union B c= union ((B \ D) \/ {x})
proof
let s be set ; :: according to TARSKI:def 3
assume E73: s in union B ;
consider t being set such that
E75: ( s in t & t in B ) by Th1, TARSKI:def 4;
per cases ( t in D or not t in D ) ;
suppose t in D ;
then t c= x by ;
hence s in union ((B \ D) \/ {x}) by , , TARSKI:def 4;
end;
suppose not t in D ;
then t in B \ D by , XBOOLE_0:def 4;
hence s in union ((B \ D) \/ {x}) by , , TARSKI:def 4;
end;
end;
end;
then E76: B = (B \ D) \/ {x} by , , ;
E77: {x} c= (B \ D) \/ {x} by XBOOLE_1:7;
x in {x} by TARSKI:def 1;
hence contradiction by , , Th1, , TARSKI:def 4;
end;
hence y in union B ;
end;
end;
end;

E92: now
let Y be set ;
let H be Hierarchy of Y;
let B be mutually-disjoint Subset-Family of Y;
assume E29: ( B c= H & ( for C being mutually-disjoint Subset-Family of Y st C c= H & union B c= union C holds
B = C ) ) ;
let S1 be Subset of Y;
assume E31: S1 in B ;
now
assume E34: S1 = {} ;
set C = B \ {{} };
E35: union B c= union (B \ {{} })
proof
let x be set ; :: according to TARSKI:def 3
assume E47: x in union B ;
consider y being set such that
E48: ( x in y & y in B ) by , TARSKI:def 4;
not y in {{} } by , TARSKI:def 1;
then y in B \ {{} } by , XBOOLE_0:def 4;
hence x in union (B \ {{} }) by , TARSKI:def 4;
end;
B \ {{} } c= B by XBOOLE_1:36;
then E49: B \ {{} } c= H by , XBOOLE_1:1;
B \ {{} } is mutually-disjoint
proof
let x be set , y be set ; :: according to TAXONOM2:def 5
assume E51: ( x in B \ {{} } & y in B \ {{} } & x <> y ) ;
( x in B & y in B ) by , XBOOLE_0:def 4;
hence x misses y by , ;
end;
then B = B \ {{} } by , , ;
then not {} in {{} } by , , XBOOLE_0:def 4;
hence contradiction by TARSKI:def 1;
end;
hence S1 <> {} ;
end;

definition
let Y be set ;
let F be Subset-Family of Y;
attr a2 is T_3 means :Def6: :: TAXONOM2:def 6
for A being Subset of Y st A in F holds
for x being Element of Y st not x in A holds
ex B being Subset of Y st
( x in B & B in F & A misses B );
end;

:: deftheorem Def6 defines T_3 TAXONOM2:def 6 :
for Y being set
for F being Subset-Family of Y holds
( F is T_3 iff for A being Subset of Y st A in F holds
for x being Element of Y st not x in A holds
ex B being Subset of Y st
( x in B & B in F & A misses B ) );

theorem Th11: :: TAXONOM2:11
for Y being set
for F being Subset-Family of Y st F = {} holds
F is T_3
proof end;

registration
let Y be set ;
cluster covering T_3 Hierarchy of a1;
existence
ex b1 being Hierarchy of Y st
( b1 is covering & b1 is T_3 )
proof end;
end;

definition
let Y be set ;
let F be Subset-Family of Y;
attr a2 is lower-bounded means :Def7: :: TAXONOM2:def 7
for B being set st B <> {} & B c= F & B is c=-linear holds
ex c being set st
( c in F & c c= meet B );
end;

:: deftheorem Def7 defines lower-bounded TAXONOM2:def 7 :
for Y being set
for F being Subset-Family of Y holds
( F is lower-bounded iff for B being set st B <> {} & B c= F & B is c=-linear holds
ex c being set st
( c in F & c c= meet B ) );

theorem Th12: :: TAXONOM2:12
for Y being set
for S1 being Subset of Y
for B being mutually-disjoint Subset-Family of Y st ( for b being 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 ) )
proof end;

definition
let Y be set ;
let F be Subset-Family of Y;
attr a2 is with_max's means :Def8: :: TAXONOM2:def 8
for S being Subset of Y st S in F holds
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;

:: deftheorem Def8 defines with_max's TAXONOM2:def 8 :
for Y being set
for F being Subset-Family of Y holds
( F is with_max's iff for S being Subset of Y st S in F holds
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 ) ) );

theorem Th13: :: TAXONOM2:13
for Y being set
for H being covering Hierarchy of Y st H is with_max's holds
ex P being a_partition of Y st P c= H
proof end;

theorem Th14: :: TAXONOM2:14
for Y being set
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
proof end;

theorem Th15: :: TAXONOM2:15
for Y being set
for H being covering T_3 Hierarchy of Y st H is lower-bounded & not {} in H holds
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
proof end;

theorem Th16: :: TAXONOM2:16
for Y being set
for H being covering T_3 Hierarchy of Y st H is lower-bounded & not {} in H holds
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
proof end;

theorem Th17: :: TAXONOM2:17
for Y being set
for H being covering T_3 Hierarchy of Y st H is lower-bounded & not {} in H holds
for A being Subset of Y st A in H holds
ex P being a_partition of Y st
( A in P & P c= H )
proof end;

E146: now
let x be non empty set ;
let y be set ;
assume E29: x c= y ;
consider a being set such that
E31: a in x by XBOOLE_0:def 1;
thus x meets y by , , XBOOLE_0:3;
end;

theorem Th18: :: TAXONOM2:18
for X, h being non empty set
for Pmin being a_partition of X
for hw being set st hw in Pmin & h c= hw holds
for PS being a_partition of X st h in PS & ( for x being set holds
( not x in PS or x c= hw or hw c= x or hw misses x ) ) holds
for PT being set st ( for a being set 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 )
proof end;

theorem Th19: :: TAXONOM2:19
for X, h being non empty set st h c= X holds
for Pmax being a_partition of X st ex hy being set st
( hy in Pmax & hy c= h ) & ( for x being set holds
( not x in Pmax or x c= h or h c= x or h misses x ) ) holds
for Pb being set st ( for x being set 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 holds
for hw being set st hw in Pmin & h c= hw holds
Pb \/ {h} is_finer_than Pmin ) )
proof end;

theorem Th20: :: TAXONOM2:20
for X being non empty set
for H being covering T_3 Hierarchy of X st H is lower-bounded & not {} in H & ( for C1 being set st C1 <> {} & C1 c= PARTITIONS X & ( for P1, P2 being set st P1 in C1 & P2 in C1 & not P1 is_finer_than P2 holds
P2 is_finer_than P1 ) holds
ex PX, PY being set st
( PX in C1 & PY in C1 & ( for PZ being set st PZ in C1 holds
( PZ is_finer_than PY & PX is_finer_than PZ ) ) ) ) holds
ex C being Classification of X st union C = H
proof end;