:: Auxiliary and Approximating Relations
::  by Adam Grabowski
::
:: Received October 21, 1996
:: Copyright (c) 1996-2021 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 XBOOLE_0, RELAT_2, ORDERS_2, RELAT_1, SUBSET_1, WAYBEL_3,
      STRUCT_0, XXREAL_0, EQREL_1, LATTICES, LATTICE3, WAYBEL_0, TARSKI,
      ZFMISC_1, YELLOW_1, SETFAM_1, REWRITE1, YELLOW_0, FUNCT_1, CARD_FIL,
      SEQM_3, PARTFUN1, CAT_1, NEWTON, FUNCOP_1, FUNCT_2, CARD_3, RLVECT_2,
      WELLORD1, YELLOW_2, WELLORD2, ORDINAL2, WAYBEL_2, ARYTM_3, ORDERS_1,
      WAYBEL_4;
 notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, SETFAM_1, RELAT_1, RELAT_2,
      FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, DOMAIN_1, ORDERS_1, STRUCT_0,
      ORDERS_2, LATTICE3, WELLORD1, YELLOW_0, FUNCOP_1, CARD_3, PRALG_1,
      YELLOW_1, ALG_1, YELLOW_2, YELLOW_4, WAYBEL_0, WAYBEL_1, WAYBEL_2,
      WAYBEL_3;
 constructors SETFAM_1, DOMAIN_1, TOLER_1, PRALG_1, ORDERS_3, WAYBEL_1,
      YELLOW_4, WAYBEL_2, WAYBEL_3, RELSET_1;
 registrations SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2, STRUCT_0, LATTICE3,
      YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_2, WAYBEL_2, WAYBEL_3, FUNCOP_1;
 requirements SUBSET, BOOLE;


begin :: Auxiliary Relations

definition
  let L be non empty reflexive RelStr;

  func L-waybelow -> Relation of L means
:: WAYBEL_4:def 1

  for x,y being Element of L holds [x,y] in it iff x << y;
end;

definition
  let L be RelStr;
  func IntRel L -> Relation of L equals
:: WAYBEL_4:def 2
  the InternalRel of L;
end;

definition
  let L be RelStr, R be Relation of L;
  attr R is auxiliary(i) means
:: WAYBEL_4:def 3

  for x, y being Element of L holds [x,y] in R implies x <= y;
  attr R is auxiliary(ii) means
:: WAYBEL_4:def 4

  for x, y, z, u being Element of L holds
  ( u <= x & [x,y] in R & y <= z implies [u,z] in R );
end;

definition
  let L be non empty RelStr, R be Relation of L;
  attr R is auxiliary(iii) means
:: WAYBEL_4:def 5

  for x, y, z being Element of L holds
  ( [x,z] in R & [y,z] in R implies [(x "\/" y),z] in R );
  attr R is auxiliary(iv) means
:: WAYBEL_4:def 6

  for x being Element of L holds [Bottom L,x] in R;
end;

:: Definition 1.9 p.43

definition
  let L be non empty RelStr, R be Relation of L;
  attr R is auxiliary means
:: WAYBEL_4:def 7

  R is auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv);
end;

registration
  let L be non empty RelStr;
  cluster auxiliary ->
    auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) for Relation of L;
  cluster auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) ->
    auxiliary for Relation of L;
end;

registration
  let L be lower-bounded with_suprema transitive antisymmetric RelStr;
  cluster auxiliary for Relation of L;
end;

theorem :: WAYBEL_4:1
  for L being lower-bounded sup-Semilattice
  for AR being auxiliary(ii) auxiliary(iii) Relation of L
  for x, y, z, u being Element of L holds
  [x, z] in AR & [y, u] in AR implies [x "\/" y, z "\/" u] in AR;

registration
  let L be lower-bounded sup-Semilattice;
  cluster auxiliary(i) auxiliary(ii) -> transitive for Relation of L;
end;

registration
  let L be RelStr;
  cluster IntRel L -> auxiliary(i);
end;

registration
  let L be transitive RelStr;
  cluster IntRel L -> auxiliary(ii);
end;

registration
  let L be with_suprema antisymmetric RelStr;
  cluster IntRel L -> auxiliary(iii);
end;

registration
  let L be lower-bounded antisymmetric non empty RelStr;
  cluster IntRel L -> auxiliary(iv);
end;

reserve a for set;

definition
  let L be lower-bounded sup-Semilattice;
  func Aux L -> set means
:: WAYBEL_4:def 8

  a in it iff a is auxiliary Relation of L;
end;

registration
  let L be lower-bounded sup-Semilattice;
  cluster Aux L -> non empty;
end;

theorem :: WAYBEL_4:2
  for L being lower-bounded sup-Semilattice
  for AR being auxiliary(i) Relation of L holds AR c= IntRel L;

theorem :: WAYBEL_4:3
  for L being lower-bounded sup-Semilattice holds
  Top InclPoset Aux L = IntRel L;

registration
  let L be lower-bounded sup-Semilattice;
  cluster InclPoset Aux L -> upper-bounded;
end;

definition
  let L be non empty RelStr;
  func AuxBottom L -> Relation of L means
:: WAYBEL_4:def 9

  for x,y be Element of L holds [x,y] in it iff x = Bottom L;
end;

registration
  let L be lower-bounded sup-Semilattice;
  cluster AuxBottom L -> auxiliary;
end;

theorem :: WAYBEL_4:4
  for L being lower-bounded sup-Semilattice
  for AR being auxiliary(iv) Relation of L holds AuxBottom L c= AR;

theorem :: WAYBEL_4:5
  for L being lower-bounded sup-Semilattice holds
  Bottom InclPoset Aux L = AuxBottom L;

registration
  let L be lower-bounded sup-Semilattice;
  cluster InclPoset Aux L -> lower-bounded;
end;

theorem :: WAYBEL_4:6
  for L being lower-bounded sup-Semilattice
  for a,b being auxiliary(i) Relation of L holds
  a /\ b is auxiliary(i) Relation of L;

theorem :: WAYBEL_4:7
  for L being lower-bounded sup-Semilattice
  for a,b being auxiliary(ii) Relation of L holds
  a /\ b is auxiliary(ii) Relation of L;

theorem :: WAYBEL_4:8
  for L being lower-bounded sup-Semilattice
  for a,b being auxiliary(iii) Relation of L holds
  a /\ b is auxiliary(iii) Relation of L;

theorem :: WAYBEL_4:9
  for L being lower-bounded sup-Semilattice
  for a,b being auxiliary(iv) Relation of L holds
  a /\ b is auxiliary(iv) Relation of L;

theorem :: WAYBEL_4:10
  for L being lower-bounded sup-Semilattice
  for a,b being auxiliary Relation of L holds
  a /\ b is auxiliary Relation of L;

theorem :: WAYBEL_4:11
  for L being lower-bounded sup-Semilattice
  for X being non empty Subset of InclPoset Aux L holds
  meet X is auxiliary Relation of L;

registration
  let L be lower-bounded sup-Semilattice;
  cluster InclPoset Aux L -> with_infima;
end;

registration
  let L be lower-bounded sup-Semilattice;
  cluster InclPoset Aux L -> complete;
end;

definition
  let L be non empty RelStr, x be Element of L;
  let AR be Relation of the carrier of L;
  func AR-below x -> Subset of L equals
:: WAYBEL_4:def 10
  {y where y is Element of L: [y,x] in AR};
  func AR-above x -> Subset of L equals
:: WAYBEL_4:def 11
  {y where y is Element of L: [x,y] in AR};
end;

theorem :: WAYBEL_4:12
  for L being lower-bounded sup-Semilattice, x being Element of L
  for AR being auxiliary(i) Relation of L holds AR-below x c= downarrow x;

registration
  let L be lower-bounded sup-Semilattice, x be Element of L;
  let AR be auxiliary(iv) Relation of L;
  cluster AR-below x -> non empty;
end;

registration
  let L be lower-bounded sup-Semilattice, x be Element of L;
  let AR be auxiliary(ii) Relation of L;
  cluster AR-below x -> lower;
end;

registration
  let L be lower-bounded sup-Semilattice, x be Element of L;
  let AR be auxiliary(iii) Relation of L;
  cluster AR-below x -> directed;
end;

definition
  let L be lower-bounded sup-Semilattice;
  let AR be auxiliary(ii) auxiliary(iii) auxiliary(iv) Relation of L;
  func AR-below -> Function of L, InclPoset Ids L means
:: WAYBEL_4:def 12

  for x be Element of L holds it.x = AR-below x;
end;

theorem :: WAYBEL_4:13
  for L being non empty RelStr, AR being Relation of L
  for a being object, y being Element of L
   holds a in AR-below y iff [a,y] in AR;

theorem :: WAYBEL_4:14
  for L being sup-Semilattice, AR being Relation of L
  for y being Element of L holds a in AR-above y iff [y,a] in AR;

theorem :: WAYBEL_4:15
  for L being lower-bounded sup-Semilattice,
  AR being auxiliary(i) Relation of L for x being Element of L holds
  AR = the InternalRel of L implies AR-below x = downarrow x;

definition
  let L be non empty Poset;
  func MonSet L -> strict RelStr means
:: WAYBEL_4:def 13

  ( a in the carrier of it iff
  ex s be Function of L, InclPoset Ids L st a = s & s is monotone &
  for x be Element of L holds s.x c= downarrow x ) &
  for c, d be object holds [c,d] in the InternalRel of it iff
  ex f,g be Function of L, InclPoset Ids L st c = f & d = g &
  c in the carrier of it & d in the carrier of it & f <= g;
end;

theorem :: WAYBEL_4:16
  for L being lower-bounded sup-Semilattice holds
  MonSet L is full SubRelStr of (InclPoset Ids L)|^(the carrier of L);

theorem :: WAYBEL_4:17
  for L being lower-bounded sup-Semilattice,
  AR being auxiliary(ii) Relation of L for x, y being Element of L holds
  x <= y implies AR-below x c= AR-below y;

registration
  let L be lower-bounded sup-Semilattice;
  let AR be auxiliary(ii) auxiliary(iii) auxiliary(iv) Relation of L;
  cluster AR-below -> monotone;
end;

theorem :: WAYBEL_4:18
  for L being lower-bounded sup-Semilattice, AR being auxiliary Relation of L
  holds AR-below in the carrier of MonSet L;

registration
  let L be lower-bounded sup-Semilattice;
  cluster MonSet L -> non empty;
end;

theorem :: WAYBEL_4:19
  for L being lower-bounded sup-Semilattice holds
  IdsMap L in the carrier of MonSet L;

theorem :: WAYBEL_4:20
  for L being lower-bounded sup-Semilattice, AR being auxiliary Relation of L
  holds AR-below <= IdsMap L;

theorem :: WAYBEL_4:21
  for L being lower-bounded non empty Poset
  for I being Ideal of L holds Bottom L in I;

theorem :: WAYBEL_4:22
  for L being upper-bounded non empty Poset
  for F being Filter of L holds Top L in F;

theorem :: WAYBEL_4:23
  for L being lower-bounded non empty Poset holds
  downarrow Bottom L = {Bottom L};

theorem :: WAYBEL_4:24
  for L being upper-bounded non empty Poset holds uparrow Top L = {Top L};

reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;

theorem :: WAYBEL_4:25
  (the carrier of L) --> {Bottom L} is Function of L, InclPoset Ids L;

theorem :: WAYBEL_4:26
  (the carrier of L) --> {Bottom L} in the carrier of MonSet L;

theorem :: WAYBEL_4:27
  for AR being auxiliary Relation of L holds
  [(the carrier of L) --> {Bottom L} , AR-below] in the InternalRel of MonSet L
;

registration
  let L;
  cluster MonSet L -> upper-bounded;
end;

definition
  let L;
  func Rel2Map L -> Function of InclPoset Aux L, MonSet L means
:: WAYBEL_4:def 14

  for AR being auxiliary Relation of L holds it.AR = AR-below;
end;

theorem :: WAYBEL_4:28
  for R1, R2 being auxiliary Relation of L st R1 c= R2 holds
  R1-below <= R2-below;

theorem :: WAYBEL_4:29
  for R1, R2 being Relation of L st R1 c= R2 holds R1-below x c= R2-below x;

registration
  let L;
  cluster Rel2Map L -> monotone;
end;

definition
  let L;
  func Map2Rel L -> Function of MonSet L, InclPoset Aux L means
:: WAYBEL_4:def 15

  for s be object st s in the carrier of MonSet L
  ex AR be auxiliary Relation of L st AR = it.s & for x,y be object holds
  [x,y] in AR iff ex x9,y9 be Element of L,
  s9 be Function of L, InclPoset Ids L st
  x9 = x & y9 = y & s9 = s & x9 in s9.y9;
end;

registration
  let L;
  cluster Map2Rel L -> monotone;
end;

theorem :: WAYBEL_4:30
  (Map2Rel L) * (Rel2Map L) = id dom (Rel2Map L);

theorem :: WAYBEL_4:31
  (Rel2Map L) * (Map2Rel L) = id (the carrier of MonSet L);

registration
  let L;
  cluster Rel2Map L -> one-to-one;
end;

:: Proposition 1.10 (i) p.43

theorem :: WAYBEL_4:32
  (Rel2Map L)" = Map2Rel L;

:: Proposition 1.10 (ii) p.43

theorem :: WAYBEL_4:33
  Rel2Map L is isomorphic;

theorem :: WAYBEL_4:34
  for L being complete LATTICE, x being Element of L holds
  meet { I where I is Ideal of L : x <= sup I } = waybelow x;

definition
  let L be Semilattice, I be Ideal of L;
  func DownMap I -> Function of L, InclPoset Ids L means
:: WAYBEL_4:def 16

  for x be Element of L holds
  ( x <= sup I implies it.x = ( downarrow x ) /\ I ) &
  ( not x <= sup I implies it.x = downarrow x );
end;

theorem :: WAYBEL_4:35
  for L being Semilattice, I being Ideal of L holds
  DownMap I in the carrier of MonSet L;

theorem :: WAYBEL_4:36
  for L being with_infima antisymmetric reflexive RelStr, x being Element of L
  for D being non empty lower Subset of L holds
  {x} "/\" D = (downarrow x) /\ D;

begin  :: Approximating Relations
:: Definition 1.11 p.44

definition
  let L be non empty RelStr, AR be Relation of L;
  attr AR is approximating means
:: WAYBEL_4:def 17

  for x be Element of L holds x = sup (AR-below x);
end;

definition
  let L be non empty Poset;
  let mp be Function of L, InclPoset Ids L;
  attr mp is approximating means
:: WAYBEL_4:def 18

  for x be Element of L ex ii be Subset of L st ii = mp.x & x = sup ii;
end;

:: Lemma 1.12 (i) p.44

theorem :: WAYBEL_4:37
  for L being lower-bounded meet-continuous Semilattice, I being Ideal of L
  holds DownMap I is approximating;

:: Lemma 1.12 (ii) p.44

theorem :: WAYBEL_4:38
  for L being lower-bounded continuous LATTICE holds L is meet-continuous;

registration
  cluster continuous -> meet-continuous for lower-bounded LATTICE;
end;

:: Lemma 1.12 (iii) p.44

theorem :: WAYBEL_4:39
  for L being lower-bounded continuous LATTICE,
  I being Ideal of L holds DownMap I is approximating;

registration
  let L be non empty reflexive antisymmetric RelStr;
  cluster L-waybelow -> auxiliary(i);
end;

registration
  let L be non empty reflexive transitive RelStr;
  cluster L-waybelow -> auxiliary(ii);
end;

registration
  let L be with_suprema Poset;
  cluster L-waybelow -> auxiliary(iii);
end;

registration
  let L be /\-complete non empty Poset;
  cluster L-waybelow -> auxiliary(iii);
end;

registration
  let L be lower-bounded antisymmetric reflexive non empty RelStr;
  cluster L-waybelow -> auxiliary(iv);
end;

theorem :: WAYBEL_4:40
  for L being complete LATTICE, x being Element of L holds
  (L-waybelow)-below x = waybelow x;

theorem :: WAYBEL_4:41
  for L being LATTICE holds IntRel L is approximating;

registration
  let L be lower-bounded continuous LATTICE;
  cluster L-waybelow -> approximating;
end;

registration
  let L be complete LATTICE;
  cluster approximating auxiliary for Relation of L;
end;

definition
  let L be complete LATTICE;
  func App L -> set means
:: WAYBEL_4:def 19

  a in it iff a is approximating auxiliary Relation of L;
end;

registration
  let L be complete LATTICE;
  cluster App L -> non empty;
end;

theorem :: WAYBEL_4:42
  for L being complete LATTICE for mp being Function of L, InclPoset Ids L st
  mp is approximating & mp in the carrier of MonSet L
  ex AR being approximating auxiliary Relation of L st AR = (Map2Rel L).mp;

theorem :: WAYBEL_4:43
  for L being complete LATTICE, x being Element of L holds
  meet the set of all  (DownMap I).x where I is Ideal of L  = waybelow x;

:: Proposition 1.13 p.45

theorem :: WAYBEL_4:44
  for L being lower-bounded meet-continuous LATTICE, x being Element of L holds
  meet {AR-below x where AR is auxiliary Relation of L : AR in App L}
  = waybelow x;

reserve L for complete LATTICE;

:: Proposition 1.14 p.45   ( 1 <=> 2 )

theorem :: WAYBEL_4:45
  L is continuous iff for R being approximating auxiliary Relation of L holds
  L-waybelow c= R & L-waybelow is approximating;

:: Proposition 1.14 p.45   ( 1 <=> 3 )

theorem :: WAYBEL_4:46
  L is continuous iff (L is meet-continuous &
  ex R being approximating auxiliary Relation of L st
  for R9 being approximating auxiliary Relation of L holds R c= R9);

:: Definition 1.15 (SI) p.46

definition
  let L be RelStr, AR be Relation of L;
  attr AR is satisfying_SI means
:: WAYBEL_4:def 20

  for x, z be Element of L holds ( [x,z] in AR & x <> z implies
  ex y be Element of L st [x,y] in AR & [y,z] in AR & x <> y );
end;

:: Definition 1.15 (INT) p.46

definition
  let L be RelStr, AR be Relation of L;
  attr AR is satisfying_INT means
:: WAYBEL_4:def 21

  for x, z be Element of L holds ( [x,z] in AR implies
  ex y be Element of L st [x,y] in AR & [y,z] in AR );
end;

theorem :: WAYBEL_4:47
  for L being RelStr, AR being Relation of L
  holds AR is satisfying_SI implies AR is satisfying_INT;

registration
  let L be non empty RelStr;
  cluster satisfying_SI -> satisfying_INT for Relation of L;
end;

reserve AR for Relation of L;
reserve x, y, z for Element of L;

theorem :: WAYBEL_4:48
  for AR being approximating Relation of L st
  not x <= y holds ex u being Element of L st [u,x] in AR & not u <= y;

:: Lemma 1.16 p.46

theorem :: WAYBEL_4:49
  for R being approximating auxiliary(i) auxiliary(iii) Relation of L holds
  [x,z] in R & x <> z implies ex y st x <= y & [y,z] in R & x <> y;

:: Lemma 1.17 p.46

theorem :: WAYBEL_4:50
  for R being approximating auxiliary Relation of L holds
  x << z & x <> z implies
  ex y being Element of L st [x,y] in R & [y,z] in R & x <> y;

:: Theorem 1.18 p.47

theorem :: WAYBEL_4:51
  for L being lower-bounded continuous LATTICE
  holds L-waybelow is satisfying_SI;

registration
  let L be lower-bounded continuous LATTICE;
  cluster L-waybelow -> satisfying_SI;
end;

theorem :: WAYBEL_4:52
  for L being lower-bounded continuous LATTICE,
  x,y being Element of L st x << y
  ex x9 being Element of L st x << x9 & x9 << y;

:: Corollary 1.19 p.47

theorem :: WAYBEL_4:53
  for L being lower-bounded continuous LATTICE holds
  for x,y being Element of L holds ( x << y iff
  for D being non empty directed Subset of L st y <= sup D
  ex d being Element of L st d in D & x << d );

begin :: Exercises

definition
  let L be RelStr, X be Subset of L, R be Relation of the carrier of L;
  pred X is_directed_wrt R means
:: WAYBEL_4:def 22

  for x,y being Element of L st x in X & y in X
  ex z being Element of L st z in X & [x,z] in R & [y,z] in R;
end;

theorem :: WAYBEL_4:54
  for L being RelStr, X being Subset of L st
  X is_directed_wrt (the InternalRel of L) holds X is directed;

definition
  let X be set, x be object;
  let R be Relation;
  pred x is_maximal_wrt X,R means
:: WAYBEL_4:def 23

  x in X & not ex y being set st y in X & y <> x & [x,y] in R;
end;

definition
  let L be RelStr, X be set, x be Element of L;
  pred x is_maximal_in X means
:: WAYBEL_4:def 24

  x is_maximal_wrt X, the InternalRel of L;
end;

theorem :: WAYBEL_4:55
  for L being RelStr, X being set, x being Element of L holds
  x is_maximal_in X iff x in X & not ex y being Element of L st y in X & x < y;

definition
  let X be set, x be object;
  let R be Relation;
  pred x is_minimal_wrt X,R means
:: WAYBEL_4:def 25

  x in X & not ex y being set st y in X & y <> x & [y,x] in R;
end;

definition
  let L be RelStr, X be set, x be Element of L;
  pred x is_minimal_in X means
:: WAYBEL_4:def 26

  x is_minimal_wrt X, the InternalRel of L;
end;

theorem :: WAYBEL_4:56
  for L being RelStr, X being set, x being Element of L holds
  x is_minimal_in X iff x in X & not ex y being Element of L st y in X & x > y;

:: Exercise 1.23 (i)    ( 1 => 2 )

theorem :: WAYBEL_4:57
  AR is satisfying_SI implies
  for x holds ( ex y st y is_maximal_wrt (AR-below x), AR )
  implies [x,x] in AR;

:: Exercise 1.23 (i)    ( 2 => 1 )

theorem :: WAYBEL_4:58
  ( for x holds ( ex y st y is_maximal_wrt (AR-below x), AR )
  implies [x,x] in AR ) implies AR is satisfying_SI;

:: Exercise 1.23 (ii)    ( 3 => 4 )

theorem :: WAYBEL_4:59
  for AR being auxiliary(ii) auxiliary(iii) Relation of L holds
  AR is satisfying_INT implies for x holds AR-below x is_directed_wrt AR;

:: Exercise 1.23 (ii)    ( 4 => 3 )

theorem :: WAYBEL_4:60
  ( for x holds AR-below x is_directed_wrt AR) implies AR is satisfying_INT;

:: Exercise 1.23 (iii) p.51

theorem :: WAYBEL_4:61
  for R being approximating auxiliary(i) auxiliary(ii) auxiliary(iii)
  Relation of L st R is satisfying_INT holds R is satisfying_SI;

registration
  let L;
  cluster satisfying_INT ->
    satisfying_SI for approximating auxiliary Relation of L;
end;