:: WAYBEL35 semantic presentation

Lm1: for x, y, X being set holds
( not y in {x} \/ X or y = x or y in X )

proof end;

registration

let X be set ;
cluster trivial Element of bool X;
existence

proof end;

end;

Lm2: for Y being trivial set
for A being Subset of Y holds A is trivial

proof end;

registration

let X be trivial set ;
cluster -> trivial Element of bool X;
coherence by Lm2;

end;

registration

let L be 1-sorted ;
cluster trivial Element of bool the carrier of L;
existence

proof end;

end;

registration

let L be RelStr ;
cluster trivial Element of bool the carrier of L;
existence

proof end;

end;

registration

let L be non empty 1-sorted ;
cluster non empty trivial Element of bool the carrier of L;
existence

proof end;

end;

registration

let L be non empty RelStr ;
cluster non empty trivial Element of bool the carrier of L;
existence

proof end;

end;

theorem Th1: :: WAYBEL35:1

for X being set holds RelIncl X is_reflexive_in X

proof end;

theorem Th2: :: WAYBEL35:2

for X being set holds RelIncl X is_transitive_in X

proof end;

theorem Th3: :: WAYBEL35:3

for X being set holds RelIncl X is_antisymmetric_in X

proof end;

registration

let L be RelStr ;
cluster auxiliary(i) Relation of the carrier of L,the carrier of L;
existence

proof end;

end;

registration

let L be transitive RelStr ;
cluster auxiliary(i) auxiliary(ii) Relation of the carrier of L,the carrier of L;
existence

proof end;

end;

registration

let L be antisymmetric with_suprema RelStr ;
cluster auxiliary(iii) Relation of the carrier of L,the carrier of L;
existence

proof end;

end;

registration

let L be non empty antisymmetric lower-bounded RelStr ;
cluster auxiliary(iv) Relation of the carrier of L,the carrier of L;
existence

proof end;

end;

definition

let L be non empty RelStr ;
let R be Relation of L;
attr R is extra-order means :Def1: :: WAYBEL35:def 1
( R is auxiliary(i) & R is auxiliary(ii) & R is auxiliary(iv) );

end;

:: deftheorem Def1 defines extra-order WAYBEL35:def 1 :

registration

let L be non empty RelStr ;
cluster extra-order -> auxiliary(i) auxiliary(ii) auxiliary(iv) Relation of the carrier of L,the carrier of L;
coherence by Def1;
cluster auxiliary(i) auxiliary(ii) auxiliary(iv) -> extra-order Relation of the carrier of L,the carrier of L;
coherence by Def1;

end;

registration

let L be non empty RelStr ;
cluster auxiliary(iii) extra-order -> auxiliary extra-order Relation of the carrier of L,the carrier of L;
coherence

proof end;

cluster auxiliary -> auxiliary(i) auxiliary(ii) auxiliary(iv) extra-order Relation of the carrier of L,the carrier of L;
coherence

proof end;

end;

registration

let L be non empty transitive antisymmetric lower-bounded RelStr ;
cluster auxiliary(i) auxiliary(ii) auxiliary(iv) extra-order Relation of the carrier of L,the carrier of L;
existence

proof end;

end;

definition

let L be lower-bounded with_suprema Poset;
let R be auxiliary(ii) Relation of L;
func R -LowerMap -> Function of L,(InclPoset (LOWER L)) means :Def2: :: WAYBEL35:def 2
for x being Element of L holds it . x = R -below x;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines -LowerMap WAYBEL35:def 2 :

registration

let L be lower-bounded with_suprema Poset;
let R be auxiliary(ii) Relation of L;
cluster R -LowerMap -> monotone ;
coherence

proof end;

end;

definition

let L be 1-sorted ;
let R be Relation of the carrier of L;
mode strict_chain of R -> Subset of L means :Def3: :: WAYBEL35:def 3
for x, y being set st x in it & y in it & not [x,y] in R & not x = y holds
[y,x] in R;
existence

proof end;

end;

:: deftheorem Def3 defines strict_chain WAYBEL35:def 3 :

theorem Th4: :: WAYBEL35:4

for L being 1-sorted
for C being trivial Subset of L
for R being Relation of the carrier of L holds C is strict_chain of R

proof end;

registration

let L be non empty 1-sorted ;
let R be Relation of the carrier of L;
cluster non empty trivial strict_chain of R;
existence

proof end;

end;

theorem Th5: :: WAYBEL35:5

for L being antisymmetric RelStr
for R being auxiliary(i) Relation of L
for C being strict_chain of R
for x, y being Element of L st x in C & y in C & x < y holds
[x,y] in R

proof end;

theorem :: WAYBEL35:6

for L being antisymmetric RelStr
for R being auxiliary(i) Relation of L
for x, y being Element of L st [x,y] in R & [y,x] in R holds
x = y

proof end;

theorem :: WAYBEL35:7

for L being non empty antisymmetric lower-bounded RelStr
for x being Element of L
for R being auxiliary(iv) Relation of L holds {(Bottom L),x} is strict_chain of R

proof end;

theorem Th8: :: WAYBEL35:8

for L being non empty antisymmetric lower-bounded RelStr
for R being auxiliary(iv) Relation of L
for C being strict_chain of R holds C \/ {(Bottom L)} is strict_chain of R

proof end;

definition

let L be 1-sorted ;
let R be Relation of the carrier of L;
let C be strict_chain of R;
attr C is maximal means :Def4: :: WAYBEL35:def 4
for D being strict_chain of R st C c= D holds
C = D;

end;

:: deftheorem Def4 defines maximal WAYBEL35:def 4 :

definition

let L be 1-sorted ;
let R be Relation of the carrier of L;
let C be set ;
func Strict_Chains R,C -> set means :Def5: :: WAYBEL35:def 5
for x being set holds
( x in it iff ( x is strict_chain of R & C c= x ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def5 defines Strict_Chains WAYBEL35:def 5 :

registration

let L be 1-sorted ;
let R be Relation of the carrier of L;
let C be strict_chain of R;
cluster Strict_Chains R,C -> non empty ;
coherence by Def5;

end;

notation

let R be Relation;
let X be set ;
synonym X is_inductive_wrt R for X has_upper_Zorn_property_wrt R;

end;

theorem :: WAYBEL35:9

for L being 1-sorted
for R being Relation of the carrier of L
for C being strict_chain of R holds
( Strict_Chains R,C is_inductive_wrt RelIncl (Strict_Chains R,C) & ex D being set st
( D is_maximal_in RelIncl (Strict_Chains R,C) & C c= D ) )

proof end;

theorem Th10: :: WAYBEL35:10

for L being non empty transitive RelStr
for C being non empty Subset of L
for X being Subset of C st ex_sup_of X,L & "\/" X,L in C holds
( ex_sup_of X, subrelstr C & "\/" X,L = "\/" X,(subrelstr C) )

proof end;

Lm3: for L being non empty Poset
for R being auxiliary(i) auxiliary(ii) Relation of L
for C being non empty strict_chain of R
for X being Subset of C st ex_sup_of X,L & C is maximal & not "\/" X,L in C holds
ex cs being Element of L st
( cs in C & "\/" X,L < cs & not [("\/" X,L),cs] in R & ex cs1 being Element of (subrelstr C) st
( cs = cs1 & X is_<=_than cs1 & ( for a being Element of (subrelstr C) st X is_<=_than a holds
cs1 <= a ) ) )

proof end;

theorem Th11: :: WAYBEL35:11

for L being non empty Poset
for R being auxiliary(i) auxiliary(ii) Relation of L
for C being non empty strict_chain of R
for X being Subset of C st ex_sup_of X,L & C is maximal holds
ex_sup_of X, subrelstr C

proof end;

theorem :: WAYBEL35:12

for L being non empty Poset
for R being auxiliary(i) auxiliary(ii) Relation of L
for C being non empty strict_chain of R
for X being Subset of C st ex_inf_of (uparrow ("\/" X,L)) /\ C,L & ex_sup_of X,L & C is maximal holds
( "\/" X,(subrelstr C) = "/\" ((uparrow ("\/" X,L)) /\ C),L & ( not "\/" X,L in C implies "\/" X,L < "/\" ((uparrow ("\/" X,L)) /\ C),L ) )

proof end;

theorem :: WAYBEL35:13

for L being non empty complete Poset
for R being auxiliary(i) auxiliary(ii) Relation of L
for C being non empty strict_chain of R st C is maximal holds
subrelstr C is complete

proof end;

theorem :: WAYBEL35:14

for L being non empty antisymmetric lower-bounded RelStr
for R being auxiliary(iv) Relation of L
for C being strict_chain of R st C is maximal holds
Bottom L in C

proof end;

theorem :: WAYBEL35:15

for L being non empty upper-bounded Poset
for R being auxiliary(ii) Relation of L
for C being strict_chain of R
for m being Element of L st C is maximal & m is_maximum_of C & [m,(Top L)] in R holds
( [(Top L),(Top L)] in R & m = Top L )

proof end;

definition

let L be RelStr ;
let C be set ;
let R be Relation of the carrier of L;
pred R satisfies_SIC_on C means :Def6: :: WAYBEL35:def 6
for x, z being Element of L st x in C & z in C & [x,z] in R & x <> z holds
ex y being Element of L st
( y in C & [x,y] in R & [y,z] in R & x <> y );

end;

:: deftheorem Def6 defines satisfies_SIC_on WAYBEL35:def 6 :

definition

let L be RelStr ;
let R be Relation of the carrier of L;
let C be strict_chain of R;
attr C is satisfying_SIC means :Def7: :: WAYBEL35:def 7
R satisfies_SIC_on C;

end;

:: deftheorem Def7 defines satisfying_SIC WAYBEL35:def 7 :

notation

let L be RelStr ;
let R be Relation of the carrier of L;
let C be strict_chain of R;
synonym satisfying_the_interpolation_property C for satisfying_SIC CC satisfies_the_interpolation_property satisfies_the_interpolation_property ;
synonym C satisfies_the_interpolation_property for satisfying_SIC CC satisfies_the_interpolation_property satisfies_the_interpolation_property ;

end;

theorem Th16: :: WAYBEL35:16

for L being RelStr
for C being set
for R being auxiliary(i) Relation of L st R satisfies_SIC_on C holds
for x, z being Element of L st x in C & z in C & [x,z] in R & x <> z holds
ex y being Element of L st
( y in C & [x,y] in R & [y,z] in R & x < y )

proof end;

registration

let L be RelStr ;
let R be Relation of the carrier of L;
cluster trivial -> satisfying_SIC strict_chain of R;
coherence

proof end;

end;

registration

let L be non empty RelStr ;
let R be Relation of the carrier of L;
cluster non empty trivial satisfying_SIC strict_chain of R;
existence

proof end;

end;

theorem :: WAYBEL35:17

for L being lower-bounded with_suprema Poset
for R being auxiliary(i) auxiliary(ii) Relation of L
for C being strict_chain of R st C is maximal & R is satisfying_SI holds
R satisfies_SIC_on C

proof end;

definition

let R be Relation;
let C, y be set ;
func SetBelow R,C,y -> set equals :: WAYBEL35:def 8
(R " {y}) /\ C;
coherence ;

end;

:: deftheorem defines SetBelow WAYBEL35:def 8 :

theorem Th18: :: WAYBEL35:18

for R being Relation
for C, x, y being set holds
( x in SetBelow R,C,y iff ( [x,y] in R & x in C ) )

proof end;

definition

let L be 1-sorted ;
let R be Relation of the carrier of L;
let C, y be set ;
:: original: SetBelow
redefine func SetBelow R,C,y -> Subset of L;
coherence

proof end;

end;

theorem Th19: :: WAYBEL35:19

for L being RelStr
for R being auxiliary(i) Relation of L
for C being set
for y being Element of L holds SetBelow R,C,y is_<=_than y

proof end;

theorem Th20: :: WAYBEL35:20

for L being reflexive transitive RelStr
for R being auxiliary(ii) Relation of L
for C being Subset of L
for x, y being Element of L st x <= y holds
SetBelow R,C,x c= SetBelow R,C,y

proof end;

theorem Th21: :: WAYBEL35:21

for L being RelStr
for R being auxiliary(i) Relation of L
for C being set
for x being Element of L st x in C & [x,x] in R & ex_sup_of SetBelow R,C,x,L holds
x = sup (SetBelow R,C,x)

proof end;

definition

let L be RelStr ;
let C be Subset of L;
attr C is sup-closed means :Def9: :: WAYBEL35:def 9
for X being Subset of C st ex_sup_of X,L holds
"\/" X,L = "\/" X,(subrelstr C);

end;

:: deftheorem Def9 defines sup-closed WAYBEL35:def 9 :

theorem Th22: :: WAYBEL35:22

for L being non empty complete Poset
for R being extra-order Relation of L
for C being satisfying_SIC strict_chain of R
for p, q being Element of L st p in C & q in C & p < q holds
ex y being Element of L st
( p < y & [y,q] in R & y = sup (SetBelow R,C,y) )

proof end;

theorem :: WAYBEL35:23

for L being non empty lower-bounded Poset
for R being extra-order Relation of L
for C being non empty strict_chain of R st C is sup-closed & ( for c being Element of L st c in C holds
ex_sup_of SetBelow R,C,c,L ) & R satisfies_SIC_on C holds
for c being Element of L st c in C holds
c = sup (SetBelow R,C,c)

proof end;

theorem :: WAYBEL35:24

for L being non empty reflexive antisymmetric RelStr
for R being auxiliary(i) Relation of L
for C being strict_chain of R st ( for c being Element of L st c in C holds
( ex_sup_of SetBelow R,C,c,L & c = sup (SetBelow R,C,c) ) ) holds
R satisfies_SIC_on C

proof end;

definition

let L be non empty RelStr ;
let R be Relation of the carrier of L;
let C be set ;
func SupBelow R,C -> set means :Def10: :: WAYBEL35:def 10
for y being set holds
( y in it iff y = sup (SetBelow R,C,y) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def10 defines SupBelow WAYBEL35:def 10 :

definition

let L be non empty RelStr ;
let R be Relation of the carrier of L;
let C be set ;
:: original: SupBelow
redefine func SupBelow R,C -> Subset of L;
coherence

proof end;

end;

theorem Th25: :: WAYBEL35:25

for L being non empty reflexive transitive RelStr
for R being auxiliary(i) auxiliary(ii) Relation of L
for C being strict_chain of R st ( for c being Element of L holds ex_sup_of SetBelow R,C,c,L ) holds
SupBelow R,C is strict_chain of R

proof end;

theorem :: WAYBEL35:26

for L being non empty Poset
for R being auxiliary(i) auxiliary(ii) Relation of L
for C being Subset of L st ( for c being Element of L holds ex_sup_of SetBelow R,C,c,L ) holds
SupBelow R,C is sup-closed

proof end;

theorem Th27: :: WAYBEL35:27

for L being non empty complete Poset
for R being extra-order Relation of L
for C being satisfying_SIC strict_chain of R
for d being Element of L st d in SupBelow R,C holds
d = "\/" { b where b is Element of L : ( b in SupBelow R,C & [b,d] in R ) } ,L

proof end;

theorem :: WAYBEL35:28

for L being non empty complete Poset
for R being extra-order Relation of L
for C being satisfying_SIC strict_chain of R holds R satisfies_SIC_on SupBelow R,C

proof end;

theorem :: WAYBEL35:29

for L being non empty complete Poset
for R being extra-order Relation of L
for C being satisfying_SIC strict_chain of R
for a, b being Element of L st a in C & b in C & a < b holds
ex d being Element of L st
( d in SupBelow R,C & a < d & [d,b] in R )

proof end;