:: Kuratowski - Zorn Lemma
:: by Wojciech A. Trybulec and Grzegorz Bancerek
::
:: Received September 19, 1989
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

attr c₁ is strict ;
struct RelStr -> 1-sorted ;
aggr RelStr(# carrier, InternalRel #) -> RelStr ;
sel InternalRel c₁ -> Relation of the carrier of c₁;

end;

registration

let X be non empty set ;
let R be Relation of X;

cluster RelStr(# X,R #) -> non empty ;

coherence
not RelStr(# X,R #) is empty ;

end;

definition

let A be RelStr ;

attr A is total means :Def1: :: ORDERS_2:def 1
the InternalRel of A is total ;

attr A is reflexive means :Def2: :: ORDERS_2:def 2
the InternalRel of A is_reflexive_in the carrier of A;

attr A is transitive means :Def3: :: ORDERS_2:def 3
the InternalRel of A is_transitive_in the carrier of A;

attr A is antisymmetric means :Def4: :: ORDERS_2:def 4
the InternalRel of A is_antisymmetric_in the carrier of A;

end;

:: deftheorem Def1 defines total ORDERS_2:def 1 :
for A being RelStr holds
( A is total iff the InternalRel of A is total );

:: deftheorem Def2 defines reflexive ORDERS_2:def 2 :
for A being RelStr holds
( A is reflexive iff the InternalRel of A is_reflexive_in the carrier of A );

:: deftheorem Def3 defines transitive ORDERS_2:def 3 :
for A being RelStr holds
( A is transitive iff the InternalRel of A is_transitive_in the carrier of A );

:: deftheorem Def4 defines antisymmetric ORDERS_2:def 4 :
for A being RelStr holds
( A is antisymmetric iff the InternalRel of A is_antisymmetric_in the carrier of A );

registration

cluster 1 -element strict total reflexive transitive antisymmetric for RelStr ;

existence
ex b₁ being RelStr st
( b₁ is reflexive & b₁ is transitive & b₁ is antisymmetric & b₁ is strict & b₁ is total & b₁ is 1 -element )

proof end;

end;

registration

cluster reflexive -> total for RelStr ;

coherence
for b₁ being RelStr st b₁ is reflexive holds
b₁ is total

proof end;

end;

definition

mode Poset is reflexive transitive antisymmetric RelStr ;

end;

registration

let A be total RelStr ;

cluster the InternalRel of A -> total ;

coherence
the InternalRel of A is total by Def1;

end;

registration

let A be reflexive RelStr ;

cluster the InternalRel of A -> reflexive ;

coherence
the InternalRel of A is reflexive

proof end;

end;

registration

let A be total antisymmetric RelStr ;

cluster the InternalRel of A -> antisymmetric ;

coherence
the InternalRel of A is antisymmetric

proof end;

end;

registration

let A be transitive RelStr ;

cluster the InternalRel of A -> transitive ;

coherence
the InternalRel of A is transitive

proof end;

end;

registration

let X be set ;
let O be reflexive total Relation of X;

cluster RelStr(# X,O #) -> reflexive ;

coherence
RelStr(# X,O #) is reflexive

proof end;

end;

registration

let X be set ;
let O be transitive total Relation of X;

cluster RelStr(# X,O #) -> transitive ;

coherence
RelStr(# X,O #) is transitive

proof end;

end;

registration

let X be set ;
let O be antisymmetric total Relation of X;

cluster RelStr(# X,O #) -> antisymmetric ;

coherence
RelStr(# X,O #) is antisymmetric

proof end;

end;

Lm1: for x being set
for A being non empty Poset
for S being Subset of A st x in S holds
x is Element of A
;

definition

let A be RelStr ;
let a1, a2 be Element of A;

pred a1 <= a2 means :: ORDERS_2:def 5
[a1,a2] in the InternalRel of A;

end;

:: deftheorem defines <= ORDERS_2:def 5 :
for A being RelStr
for a1, a2 being Element of A holds
( a1 <= a2 iff [a1,a2] in the InternalRel of A );

notation

let A be RelStr ;
let a1, a2 be Element of A;
synonym a2 >= a1 for a1 <= a2;

end;

definition

let A be RelStr ;
let a1, a2 be Element of A;

pred a1 < a2 means :: ORDERS_2:def 6
( a1 <= a2 & a1 <> a2 );

irreflexivity
for a1 being Element of A holds
( not a1 <= a1 or not a1 <> a1 ) ;

end;

:: deftheorem defines < ORDERS_2:def 6 :
for A being RelStr
for a1, a2 being Element of A holds
( a1 < a2 iff ( a1 <= a2 & a1 <> a2 ) );

notation

let A be RelStr ;
let a1, a2 be Element of A;
synonym a2 > a1 for a1 < a2;

end;

theorem Th1: :: ORDERS_2:1

for A being non empty reflexive RelStr
for a being Element of A holds a <= a

proof end;

definition

let A be non empty reflexive RelStr ;
let a1, a2 be Element of A;
:: original: <=
redefine pred a1 <= a2;
reflexivity
for a1 being Element of A holds (A,b₁,b₁) by Th1;

end;

theorem Th2: :: ORDERS_2:2

for A being antisymmetric RelStr
for a1, a2 being Element of A st a1 <= a2 & a2 <= a1 holds
a1 = a2

proof end;

theorem Th3: :: ORDERS_2:3

for A being transitive RelStr
for a1, a2, a3 being Element of A st a1 <= a2 & a2 <= a3 holds
a1 <= a3

proof end;

theorem Th4: :: ORDERS_2:4

for A being antisymmetric RelStr
for a1, a2 being Element of A holds
( not a1 < a2 or not a2 < a1 ) by Th2;

theorem Th5: :: ORDERS_2:5

for A being transitive antisymmetric RelStr
for a1, a2, a3 being Element of A st a1 < a2 & a2 < a3 holds
a1 < a3 by Th3, Th4;

theorem Th6: :: ORDERS_2:6

for A being antisymmetric RelStr
for a1, a2 being Element of A st a1 <= a2 holds
not a2 < a1 by Th2;

theorem Th7: :: ORDERS_2:7

for A being transitive antisymmetric RelStr
for a1, a2, a3 being Element of A st ( ( a1 < a2 & a2 <= a3 ) or ( a1 <= a2 & a2 < a3 ) ) holds
a1 < a3 by Th2, Th3;

::
:: Chains.
::

definition

let A be RelStr ;
let IT be Subset of A;

attr IT is strongly_connected means :Def7: :: ORDERS_2:def 7
the InternalRel of A is_strongly_connected_in IT;

end;

:: deftheorem Def7 defines strongly_connected ORDERS_2:def 7 :
for A being RelStr
for IT being Subset of A holds
( IT is strongly_connected iff the InternalRel of A is_strongly_connected_in IT );

registration

let A be RelStr ;

cluster empty -> strongly_connected for Element of bool the carrier of A;

coherence
for b₁ being Subset of A st b₁ is empty holds
b₁ is strongly_connected

proof end;

end;

registration

let A be RelStr ;

cluster strongly_connected for Element of bool the carrier of A;

existence
ex b₁ being Subset of A st b₁ is strongly_connected

proof end;

end;

definition

let A be RelStr ;
mode Chain of A is strongly_connected Subset of A;

end;

theorem Th8: :: ORDERS_2:8

for A being non empty reflexive RelStr
for a being Element of A holds {a} is Chain of A

proof end;

theorem Th9: :: ORDERS_2:9

for A being non empty reflexive RelStr
for a1, a2 being Element of A holds
( {a1,a2} is Chain of A iff ( a1 <= a2 or a2 <= a1 ) )

proof end;

theorem :: ORDERS_2:10

for A being RelStr
for C being Chain of A
for S being Subset of A st S c= C holds
S is Chain of A

proof end;

theorem Th11: :: ORDERS_2:11

for A being reflexive RelStr
for a1, a2 being Element of A holds
( ( a1 <= a2 or a2 <= a1 ) iff ex C being Chain of A st
( a1 in C & a2 in C ) )

proof end;

theorem Th12: :: ORDERS_2:12

for A being reflexive antisymmetric RelStr
for a1, a2 being Element of A holds
( ex C being Chain of A st
( a1 in C & a2 in C ) iff ( a1 < a2 iff not a2 <= a1 ) ) by Th6, Th11;

theorem Th13: :: ORDERS_2:13

for A being RelStr
for T being Subset of A st the InternalRel of A well_orders T holds
T is Chain of A by ORDERS_1:7, Def7;

::
:: Upper and lower cones.
::

definition

let A be non empty Poset;
let S be Subset of A;

func UpperCone S -> Subset of A equals :: ORDERS_2:def 8
{ a1 where a1 is Element of A : for a2 being Element of A st a2 in S holds
a2 < a1 } ;

coherence
{ a1 where a1 is Element of A : for a2 being Element of A st a2 in S holds
a2 < a1 } is Subset of A

proof end;

end;

:: deftheorem defines UpperCone ORDERS_2:def 8 :
for A being non empty Poset
for S being Subset of A holds UpperCone S = { a1 where a1 is Element of A : for a2 being Element of A st a2 in S holds
a2 < a1 } ;

definition

let A be non empty Poset;
let S be Subset of A;

func LowerCone S -> Subset of A equals :: ORDERS_2:def 9
{ a1 where a1 is Element of A : for a2 being Element of A st a2 in S holds
a1 < a2 } ;

coherence
{ a1 where a1 is Element of A : for a2 being Element of A st a2 in S holds
a1 < a2 } is Subset of A

proof end;

end;

:: deftheorem defines LowerCone ORDERS_2:def 9 :
for A being non empty Poset
for S being Subset of A holds LowerCone S = { a1 where a1 is Element of A : for a2 being Element of A st a2 in S holds
a1 < a2 } ;

theorem Th14: :: ORDERS_2:14

for A being non empty Poset holds UpperCone ({} A) = the carrier of A

proof end;

theorem :: ORDERS_2:15

for A being non empty Poset holds UpperCone ([#] A) = {}

proof end;

theorem :: ORDERS_2:16

for A being non empty Poset holds LowerCone ({} A) = the carrier of A

proof end;

theorem :: ORDERS_2:17

for A being non empty Poset holds LowerCone ([#] A) = {}

proof end;

theorem Th18: :: ORDERS_2:18

for A being non empty Poset
for a being Element of A
for S being Subset of A st a in S holds
not a in UpperCone S

proof end;

theorem :: ORDERS_2:19

for A being non empty Poset
for a being Element of A holds not a in UpperCone {a}

proof end;

theorem Th20: :: ORDERS_2:20

for A being non empty Poset
for a being Element of A
for S being Subset of A st a in S holds
not a in LowerCone S

proof end;

theorem Th21: :: ORDERS_2:21

for A being non empty Poset
for a being Element of A holds not a in LowerCone {a}

proof end;

theorem :: ORDERS_2:22

for A being non empty Poset
for a, c being Element of A holds
( c < a iff a in UpperCone {c} )

proof end;

theorem Th23: :: ORDERS_2:23

for A being non empty Poset
for a, c being Element of A holds
( a < c iff a in LowerCone {c} )

proof end;

::
:: Initial segments.
::

definition

let A be non empty Poset;
let S be Subset of A;
let a be Element of A;

func InitSegm (S,a) -> Subset of A equals :: ORDERS_2:def 10
(LowerCone {a}) /\ S;

correctness
coherence
(LowerCone {a}) /\ S is Subset of A;
;

end;

:: deftheorem defines InitSegm ORDERS_2:def 10 :
for A being non empty Poset
for S being Subset of A
for a being Element of A holds InitSegm (S,a) = (LowerCone {a}) /\ S;

definition

let A be non empty Poset;
let S be Subset of A;

mode Initial_Segm of S -> Subset of A means :Def11: :: ORDERS_2:def 11
ex a being Element of A st
( a in S & it = InitSegm (S,a) ) if S <> {}
otherwise it = {} ;

existence
( ( S <> {} implies ex b₁ being Subset of A ex a being Element of A st
( a in S & b₁ = InitSegm (S,a) ) ) & ( not S <> {} implies ex b₁ being Subset of A st b₁ = {} ) )

proof end;

consistency
for b₁ being Subset of A holds verum ;

end;

:: deftheorem Def11 defines Initial_Segm ORDERS_2:def 11 :
for A being non empty Poset
for S, b₃ being Subset of A holds
( ( S <> {} implies ( b₃ is Initial_Segm of S iff ex a being Element of A st
( a in S & b₃ = InitSegm (S,a) ) ) ) & ( not S <> {} implies ( b₃ is Initial_Segm of S iff b₃ = {} ) ) );

theorem Th24: :: ORDERS_2:24

for A being non empty Poset
for a, b being Element of A
for S being Subset of A holds
( a in InitSegm (S,b) iff ( a < b & a in S ) )

proof end;

theorem :: ORDERS_2:25

for A being non empty Poset
for a being Element of A holds InitSegm (({} A),a) = {} ;

theorem Th26: :: ORDERS_2:26

for A being non empty Poset
for a being Element of A
for S being Subset of A holds not a in InitSegm (S,a)

proof end;

theorem :: ORDERS_2:27

for A being non empty Poset
for a1, a2 being Element of A
for S being Subset of A st a1 < a2 holds
InitSegm (S,a1) c= InitSegm (S,a2)

proof end;

theorem Th28: :: ORDERS_2:28

for A being non empty Poset
for a being Element of A
for S, T being Subset of A st S c= T holds
InitSegm (S,a) c= InitSegm (T,a)

proof end;

theorem Th29: :: ORDERS_2:29

for A being non empty Poset
for S being Subset of A
for I being Initial_Segm of S holds I c= S

proof end;

theorem Th30: :: ORDERS_2:30

for A being non empty Poset
for S being Subset of A holds
( S <> {} iff not S is Initial_Segm of S )

proof end;

theorem Th31: :: ORDERS_2:31

for A being non empty Poset
for S, T being Subset of A st S <> {} & S is Initial_Segm of T holds
not T is Initial_Segm of S

proof end;

theorem Th32: :: ORDERS_2:32

for A being non empty Poset
for a1, a2 being Element of A
for S, T being Subset of A st a1 < a2 & a1 in S & a2 in T & T is Initial_Segm of S holds
a1 in T

proof end;

theorem Th33: :: ORDERS_2:33

for A being non empty Poset
for a being Element of A
for S, T being Subset of A st a in S & S is Initial_Segm of T holds
InitSegm (S,a) = InitSegm (T,a)

proof end;

theorem :: ORDERS_2:34

for A being non empty Poset
for S, T being Subset of A st S c= T & the InternalRel of A well_orders T & ( for a1, a2 being Element of A st a2 in S & a1 < a2 holds
a1 in S ) & not S = T holds
S is Initial_Segm of T

proof end;

theorem Th35: :: ORDERS_2:35

for A being non empty Poset
for S, T being Subset of A st S c= T & the InternalRel of A well_orders T & ( for a1, a2 being Element of A st a2 in S & a1 in T & a1 < a2 holds
a1 in S ) & not S = T holds
S is Initial_Segm of T

proof end;

definition

let A be non empty Poset;
let f be Choice_Function of (BOOL the carrier of A);

mode Chain of f -> Chain of A means :Def12: :: ORDERS_2:def 12
( it <> {} & the InternalRel of A well_orders it & ( for a being Element of A st a in it holds
f . (UpperCone (InitSegm (it,a))) = a ) );

existence
ex b₁ being Chain of A st
( b₁ <> {} & the InternalRel of A well_orders b₁ & ( for a being Element of A st a in b₁ holds
f . (UpperCone (InitSegm (b₁,a))) = a ) )

proof end;

end;

:: deftheorem Def12 defines Chain ORDERS_2:def 12 :
for A being non empty Poset
for f being Choice_Function of (BOOL the carrier of A)
for b₃ being Chain of A holds
( b₃ is Chain of f iff ( b₃ <> {} & the InternalRel of A well_orders b₃ & ( for a being Element of A st a in b₃ holds
f . (UpperCone (InitSegm (b₃,a))) = a ) ) );

theorem Th36: :: ORDERS_2:36

for A being non empty Poset
for f being Choice_Function of (BOOL the carrier of A) holds {(f . the carrier of A)} is Chain of f

proof end;

theorem Th37: :: ORDERS_2:37

for A being non empty Poset
for f being Choice_Function of (BOOL the carrier of A)
for fC being Chain of f holds f . the carrier of A in fC

proof end;

theorem Th38: :: ORDERS_2:38

for A being non empty Poset
for a, b being Element of A
for f being Choice_Function of (BOOL the carrier of A)
for fC being Chain of f st a in fC & b = f . the carrier of A holds
b <= a

proof end;

theorem :: ORDERS_2:39

for A being non empty Poset
for a being Element of A
for f being Choice_Function of (BOOL the carrier of A)
for fC being Chain of f st a = f . the carrier of A holds
InitSegm (fC,a) = {}

proof end;

theorem :: ORDERS_2:40

for A being non empty Poset
for f being Choice_Function of (BOOL the carrier of A)
for fC1, fC2 being Chain of f holds fC1 meets fC2

proof end;

theorem Th41: :: ORDERS_2:41

for A being non empty Poset
for f being Choice_Function of (BOOL the carrier of A)
for fC1, fC2 being Chain of f st fC1 <> fC2 holds
( fC1 is Initial_Segm of fC2 iff fC2 is not Initial_Segm of fC1 )

proof end;

theorem Th42: :: ORDERS_2:42

for A being non empty Poset
for f being Choice_Function of (BOOL the carrier of A)
for fC1, fC2 being Chain of f holds
( fC1 c< fC2 iff fC1 is Initial_Segm of fC2 )

proof end;

definition

let A be non empty Poset;
let f be Choice_Function of (BOOL the carrier of A);

func Chains f -> set means :Def13: :: ORDERS_2:def 13
for x being set holds
( x in it iff x is Chain of f );

existence
ex b₁ being set st
for x being set holds
( x in b₁ iff x is Chain of f )

proof end;

uniqueness
for b₁, b₂ being set st ( for x being set holds
( x in b₁ iff x is Chain of f ) ) & ( for x being set holds
( x in b₂ iff x is Chain of f ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def13 defines Chains ORDERS_2:def 13 :
for A being non empty Poset
for f being Choice_Function of (BOOL the carrier of A)
for b₃ being set holds
( b₃ = Chains f iff for x being set holds
( x in b₃ iff x is Chain of f ) );

registration

let A be non empty Poset;
let f be Choice_Function of (BOOL the carrier of A);

cluster Chains f -> non empty ;

coherence
not Chains f is empty

proof end;

end;

Lm2: for A being non empty Poset
for f being Choice_Function of (BOOL the carrier of A) holds union (Chains f) is Subset of A

proof end;

Lm3: for A being non empty Poset
for f being Choice_Function of (BOOL the carrier of A) holds union (Chains f) is Chain of A

proof end;

theorem Th43: :: ORDERS_2:43

for A being non empty Poset
for f being Choice_Function of (BOOL the carrier of A) holds union (Chains f) <> {}

proof end;

theorem Th44: :: ORDERS_2:44

for A being non empty Poset
for S being Subset of A
for f being Choice_Function of (BOOL the carrier of A)
for fC being Chain of f st fC <> union (Chains f) & S = union (Chains f) holds
fC is Initial_Segm of S

proof end;

theorem Th45: :: ORDERS_2:45

for A being non empty Poset
for f being Choice_Function of (BOOL the carrier of A) holds union (Chains f) is Chain of f

proof end;

::
:: Orders.
::

theorem Th46: :: ORDERS_2:46

for A being non empty Poset
for S being Subset of A holds field ( the InternalRel of A |_2 S) = S

proof end;

theorem :: ORDERS_2:47

for A being non empty Poset
for S being Subset of A st the InternalRel of A |_2 S is being_linear-order holds
S is Chain of A

proof end;

theorem :: ORDERS_2:48

for A being non empty Poset
for C being Chain of A holds the InternalRel of A |_2 C is being_linear-order

proof end;

theorem :: ORDERS_2:49

for A being non empty Poset
for S being Subset of A st the InternalRel of A linearly_orders S holds
S is Chain of A by ORDERS_1:7, Def7;

theorem :: ORDERS_2:50

for A being non empty Poset
for C being Chain of A holds the InternalRel of A linearly_orders C

proof end;

theorem :: ORDERS_2:51

for A being non empty Poset
for a being Element of A holds
( a is_minimal_in the InternalRel of A iff for b being Element of A holds not b < a )

proof end;

theorem :: ORDERS_2:52

for A being non empty Poset
for a being Element of A holds
( a is_maximal_in the InternalRel of A iff for b being Element of A holds not a < b )

proof end;

theorem :: ORDERS_2:53

for A being non empty Poset
for a being Element of A holds
( a is_superior_of the InternalRel of A iff for b being Element of A st a <> b holds
b < a )

proof end;

theorem :: ORDERS_2:54

for A being non empty Poset
for a being Element of A holds
( a is_inferior_of the InternalRel of A iff for b being Element of A st a <> b holds
a < b )

proof end;

Lm4: for X, Y being set
for R being Relation st R well_orders X & Y c= X holds
R well_orders Y

proof end;

::
:: Kuratowski - Zorn Lemma.
::

theorem Th55: :: ORDERS_2:55

for A being non empty Poset st ( for C being Chain of A ex a being Element of A st
for b being Element of A st b in C holds
b <= a ) holds
ex a being Element of A st
for b being Element of A holds not a < b

proof end;

theorem :: ORDERS_2:56

for A being non empty Poset st ( for C being Chain of A ex a being Element of A st
for b being Element of A st b in C holds
a <= b ) holds
ex a being Element of A st
for b being Element of A holds not b < a

proof end;

:: from YELLOW14, 2009.07.26, A.T.

registration

cluster empty strict for RelStr ;

existence
ex b₁ being RelStr st
( b₁ is strict & b₁ is empty )

proof end;

end;