:: On the Many Sorted Closure Operator and the Many Sorted Closure System
:: by Artur Korni{\l}owicz
::
:: Received February 7, 1996
:: Copyright (c) 1996-2021 Association of Mizar Users

scheme :: CLOSURE1:sch 1
MSSUBSET{ F₁() -> set , F₂() -> V8() ManySortedSet of F₁(), F₃() -> ManySortedSet of F₁(), P₁[ set ] } :

( ( for X being ManySortedSet of F₁() holds
( X in F₂() iff ( X in F₃() & P₁[X] ) ) ) implies F₂() c= F₃() )

proof end;

theorem Th1: :: CLOSURE1:1

for X being non empty set
for x, y being set st x c= y holds
{ t where t is Element of X : y c= t } c= { z where z is Element of X : x c= z }

proof end;

theorem :: CLOSURE1:2

for I being set
for M being ManySortedSet of I st ex A being ManySortedSet of I st A in M holds
M is V8() ;

registration

let I be set ;
let F be ManySortedFunction of I;
let A be ManySortedSet of I;

cluster F .. A -> I -defined total for I -defined Function;

coherence
for b₁ being I -defined Function st b₁ = F .. A holds
b₁ is total ;

end;

Lm1: now :: thesis: for I being set
for A, B being V8() ManySortedSet of I
for F being ManySortedFunction of A,B
for X being Element of A holds F .. X is Element of B

let I be set ; :: thesis: for A, B being V8() ManySortedSet of I
for F being ManySortedFunction of A,B
for X being Element of A holds F .. X is Element of B
let A, B be V8() ManySortedSet of I; :: thesis: for F being ManySortedFunction of A,B
for X being Element of A holds F .. X is Element of B
let F be ManySortedFunction of A,B; :: thesis: for X being Element of A holds F .. X is Element of B
let X be Element of A; :: thesis: F .. X is Element of B
thus F .. X is Element of B :: thesis: verum

proof

let i be object ; :: according to PBOOLE:def 14 :: thesis: ( not i in I or (F .. X) . i is Element of B . i )
assume A1: i in I ; :: thesis: (F .. X) . i is Element of B . i
reconsider g = F . i as Function ;
( g is Function of (A . i),(B . i) & X . i is Element of A . i ) by A1, PBOOLE:def 14, PBOOLE:def 15;
then ( dom F = I & g . (X . i) in B . i ) by A1, FUNCT_2:5, PARTFUN1:def 2;
hence (F .. X) . i is Element of B . i by A1, PRALG_1:def 20; :: thesis: verum

end;

end;

definition

let I be set ;
let A, B be V8() ManySortedSet of I;
let F be ManySortedFunction of A,B;
let X be Element of A;
:: original: ..
redefine func F .. X -> Element of B;
coherence
F .. X is Element of B by Lm1;

end;

theorem :: CLOSURE1:3

for I being set
for A, X being ManySortedSet of I
for B being V8() ManySortedSet of I
for F being ManySortedFunction of A,B st X in A holds
F .. X in B

proof end;

theorem Th4: :: CLOSURE1:4

for I being set
for F, G being ManySortedFunction of I
for A being ManySortedSet of I st A in doms G holds
F .. (G .. A) = (F ** G) .. A

proof end;

theorem :: CLOSURE1:5

for I being set
for F being ManySortedFunction of I st F is "1-1" holds
for A, B being ManySortedSet of I st A in doms F & B in doms F & F .. A = F .. B holds
A = B

proof end;

theorem :: CLOSURE1:6

for I being set
for F being ManySortedFunction of I st doms F is V8() & ( for A, B being ManySortedSet of I st A in doms F & B in doms F & F .. A = F .. B holds
A = B ) holds
F is "1-1"

proof end;

theorem Th7: :: CLOSURE1:7

for I being set
for A, B being V8() ManySortedSet of I
for F, G being ManySortedFunction of A,B st ( for M being ManySortedSet of I st M in A holds
F .. M = G .. M ) holds
F = G

proof end;

registration

let I be set ;
let M be ManySortedSet of I;

cluster Relation-like V9() I -defined Function-like total V45() for Element of bool M;

existence
ex b₁ being Element of bool M st
( b₁ is V9() & b₁ is V45() )

proof end;

end;

definition

let I be set ;
let M be ManySortedSet of I;
mode MSSetOp of M is ManySortedFunction of bool M, bool M;

end;

definition

let I be set ;
let M be ManySortedSet of I;
let O be MSSetOp of M;
let X be Element of bool M;
:: original: ..
redefine func O .. X -> Element of bool M;
coherence
O .. X is Element of bool M by Lm1;

end;

definition

let I be set ;
let M be ManySortedSet of I;
let IT be MSSetOp of M;

attr IT is reflexive means :Def1: :: CLOSURE1:def 1
for X being Element of bool M holds X c= IT .. X;

attr IT is monotonic means :: CLOSURE1:def 2
for X, Y being Element of bool M st X c= Y holds
IT .. X c= IT .. Y;

attr IT is idempotent means :: CLOSURE1:def 3
for X being Element of bool M holds IT .. X = IT .. (IT .. X);

attr IT is topological means :: CLOSURE1:def 4
for X, Y being Element of bool M holds IT .. (X (\/) Y) = (IT .. X) (\/) (IT .. Y);

end;

:: deftheorem Def1 defines reflexive CLOSURE1:def 1 :
for I being set
for M being ManySortedSet of I
for IT being MSSetOp of M holds
( IT is reflexive iff for X being Element of bool M holds X c= IT .. X );

:: deftheorem defines monotonic CLOSURE1:def 2 :
for I being set
for M being ManySortedSet of I
for IT being MSSetOp of M holds
( IT is monotonic iff for X, Y being Element of bool M st X c= Y holds
IT .. X c= IT .. Y );

:: deftheorem defines idempotent CLOSURE1:def 3 :
for I being set
for M being ManySortedSet of I
for IT being MSSetOp of M holds
( IT is idempotent iff for X being Element of bool M holds IT .. X = IT .. (IT .. X) );

:: deftheorem defines topological CLOSURE1:def 4 :
for I being set
for M being ManySortedSet of I
for IT being MSSetOp of M holds
( IT is topological iff for X, Y being Element of bool M holds IT .. (X (\/) Y) = (IT .. X) (\/) (IT .. Y) );

theorem Th8: :: CLOSURE1:8

for I being set
for M being V8() ManySortedSet of I
for X being Element of M holds X = (id M) .. X

proof end;

theorem Th9: :: CLOSURE1:9

for I being set
for M being V8() ManySortedSet of I
for X, Y being Element of M st X c= Y holds
(id M) .. X c= (id M) .. Y

proof end;

theorem Th10: :: CLOSURE1:10

for I being set
for M being V8() ManySortedSet of I
for X, Y being Element of M st X (\/) Y is Element of M holds
(id M) .. (X (\/) Y) = ((id M) .. X) (\/) ((id M) .. Y)

proof end;

theorem :: CLOSURE1:11

for I being set
for M being ManySortedSet of I
for X being Element of bool M
for i, x being set st i in I & x in ((id (bool M)) .. X) . i holds
ex Y being V45() Element of bool M st
( Y c= X & x in ((id (bool M)) .. Y) . i )

proof end;

registration

let I be set ;
let M be ManySortedSet of I;

cluster Relation-like I -defined Function-like total Function-yielding V25() reflexive monotonic idempotent topological for ManySortedFunction of bool M, bool M;

existence
ex b₁ being MSSetOp of M st
( b₁ is reflexive & b₁ is monotonic & b₁ is idempotent & b₁ is topological )

proof end;

end;

theorem :: CLOSURE1:12

for I being set
for A being ManySortedSet of I holds id (bool A) is reflexive MSSetOp of A

proof end;

theorem :: CLOSURE1:13

for I being set
for A being ManySortedSet of I holds id (bool A) is monotonic MSSetOp of A

proof end;

theorem :: CLOSURE1:14

for I being set
for A being ManySortedSet of I holds id (bool A) is idempotent MSSetOp of A

proof end;

theorem :: CLOSURE1:15

for I being set
for A being ManySortedSet of I holds id (bool A) is topological MSSetOp of A

proof end;

theorem :: CLOSURE1:16

for I being set
for M being ManySortedSet of I
for P being MSSetOp of M
for E being Element of bool M st E = M & P is reflexive holds
E = P .. E

proof end;

theorem :: CLOSURE1:17

for I being set
for M being ManySortedSet of I
for P being MSSetOp of M st P is reflexive & ( for X being Element of bool M holds P .. X c= X ) holds
P is idempotent

proof end;

theorem :: CLOSURE1:18

for I being set
for M being ManySortedSet of I
for P being MSSetOp of M
for E, T being Element of bool M st P is monotonic holds
P .. (E (/\) T) c= (P .. E) (/\) (P .. T)

proof end;

registration

let I be set ;
let M be ManySortedSet of I;

cluster topological -> monotonic for ManySortedFunction of bool M, bool M;

coherence
for b₁ being MSSetOp of M st b₁ is topological holds
b₁ is monotonic

proof end;

end;

theorem :: CLOSURE1:19

for I being set
for M being ManySortedSet of I
for P being MSSetOp of M
for E, T being Element of bool M st P is topological holds
(P .. E) (\) (P .. T) c= P .. (E (\) T)

proof end;

definition

let I be set ;
let M be ManySortedSet of I;
let R, P be MSSetOp of M;
:: original: **
redefine func P ** R -> MSSetOp of M;
coherence
P ** R is MSSetOp of M

proof end;

end;

theorem :: CLOSURE1:20

for I being set
for M being ManySortedSet of I
for P, R being MSSetOp of M st P is reflexive & R is reflexive holds
P ** R is reflexive

proof end;

theorem :: CLOSURE1:21

for I being set
for M being ManySortedSet of I
for P, R being MSSetOp of M st P is monotonic & R is monotonic holds
P ** R is monotonic

proof end;

theorem :: CLOSURE1:22

for I being set
for M being ManySortedSet of I
for P, R being MSSetOp of M st P is idempotent & R is idempotent & P ** R = R ** P holds
P ** R is idempotent

proof end;

theorem :: CLOSURE1:23

for I being set
for M being ManySortedSet of I
for P, R being MSSetOp of M st P is topological & R is topological holds
P ** R is topological

proof end;

Lm2: now :: thesis: for I being set
for M being ManySortedSet of I
for i being set
for a being ManySortedSet of I
for b being Element of bool (M . i) st a = (EmptyMS I) +* (i .--> b) holds
a in bool M

let I be set ; :: thesis: for M being ManySortedSet of I
for i being set
for a being ManySortedSet of I
for b being Element of bool (M . i) st a = (EmptyMS I) +* (i .--> b) holds
a in bool M
let M be ManySortedSet of I; :: thesis: for i being set
for a being ManySortedSet of I
for b being Element of bool (M . i) st a = (EmptyMS I) +* (i .--> b) holds
a in bool M
let i be set ; :: thesis: for a being ManySortedSet of I
for b being Element of bool (M . i) st a = (EmptyMS I) +* (i .--> b) holds
a in bool M
let a be ManySortedSet of I; :: thesis: for b being Element of bool (M . i) st a = (EmptyMS I) +* (i .--> b) holds
a in bool M
let b be Element of bool (M . i); :: thesis: ( a = (EmptyMS I) +* (i .--> b) implies a in bool M )
assume A1: a = (EmptyMS I) +* (i .--> b) ; :: thesis: a in bool M
A2: dom (i .--> b) = {i} ;
EmptyMS I c= M by MBOOLEAN:5;
then A3: EmptyMS I in bool M by MBOOLEAN:1;
thus a in bool M :: thesis: verum

proof

let j be object ; :: according to PBOOLE:def 1 :: thesis: ( not j in I or a . j in (bool M) . j )
assume A4: j in I ; :: thesis: a . j in (bool M) . j
i in {i} by TARSKI:def 1;
then A5: a . i = (i .--> b) . i by A1, A2, FUNCT_4:13
.= b by FUNCOP_1:72 ;

now :: thesis: ( ( j = i & a . j in (bool M) . j ) or ( j <> i & a . j in (bool M) . j ) )

per cases ( j = i or j <> i ) ;

case A6: j = i ; :: thesis: a . j in (bool M) . j

then b in bool (M . j) ;
hence a . j in (bool M) . j by A4, A5, A6, MBOOLEAN:def 1; :: thesis: verum

end;

case j <> i ; :: thesis: a . j in (bool M) . j

then not j in dom (i .--> b) by TARSKI:def 1;
then a . j = (EmptyMS I) . j by A1, FUNCT_4:11;
hence a . j in (bool M) . j by A3, A4; :: thesis: verum

end;

end;

end;

hence a . j in (bool M) . j ; :: thesis: verum

end;

end;

theorem Th24: :: CLOSURE1:24

for i, I being set
for M being ManySortedSet of I
for f being Function
for P being MSSetOp of M st P is reflexive & i in I & f = P . i holds
for x being Element of bool (M . i) holds x c= f . x

proof end;

theorem Th25: :: CLOSURE1:25

for i, I being set
for M being ManySortedSet of I
for f being Function
for P being MSSetOp of M st P is monotonic & i in I & f = P . i holds
for x, y being Element of bool (M . i) st x c= y holds
f . x c= f . y

proof end;

theorem Th26: :: CLOSURE1:26

for i, I being set
for M being ManySortedSet of I
for f being Function
for P being MSSetOp of M st P is idempotent & i in I & f = P . i holds
for x being Element of bool (M . i) holds f . x = f . (f . x)

proof end;

theorem :: CLOSURE1:27

for i, I being set
for M being ManySortedSet of I
for f being Function
for P being MSSetOp of M st P is topological & i in I & f = P . i holds
for x, y being Element of bool (M . i) holds f . (x \/ y) = (f . x) \/ (f . y)

proof end;

definition

let S be 1-sorted ;
attr c₂ is strict ;
struct MSClosureStr over S -> many-sorted over S;
aggr MSClosureStr(# Sorts, Family #) -> MSClosureStr over S;
sel Family c₂ -> MSSubsetFamily of the Sorts of c₂;

end;

definition

let S be 1-sorted ;
let IT be MSClosureStr over S;

attr IT is additive means :Def5: :: CLOSURE1:def 5
the Family of IT is additive ;

attr IT is absolutely-additive means :Def6: :: CLOSURE1:def 6
the Family of IT is absolutely-additive ;

attr IT is multiplicative means :Def7: :: CLOSURE1:def 7
the Family of IT is multiplicative ;

attr IT is absolutely-multiplicative means :Def8: :: CLOSURE1:def 8
the Family of IT is absolutely-multiplicative ;

attr IT is properly-upper-bound means :Def9: :: CLOSURE1:def 9
the Family of IT is properly-upper-bound ;

attr IT is properly-lower-bound means :Def10: :: CLOSURE1:def 10
the Family of IT is properly-lower-bound ;

end;

:: deftheorem Def5 defines additive CLOSURE1:def 5 :
for S being 1-sorted
for IT being MSClosureStr over S holds
( IT is additive iff the Family of IT is additive );

:: deftheorem Def6 defines absolutely-additive CLOSURE1:def 6 :
for S being 1-sorted
for IT being MSClosureStr over S holds
( IT is absolutely-additive iff the Family of IT is absolutely-additive );

:: deftheorem Def7 defines multiplicative CLOSURE1:def 7 :
for S being 1-sorted
for IT being MSClosureStr over S holds
( IT is multiplicative iff the Family of IT is multiplicative );

:: deftheorem Def8 defines absolutely-multiplicative CLOSURE1:def 8 :
for S being 1-sorted
for IT being MSClosureStr over S holds
( IT is absolutely-multiplicative iff the Family of IT is absolutely-multiplicative );

:: deftheorem Def9 defines properly-upper-bound CLOSURE1:def 9 :
for S being 1-sorted
for IT being MSClosureStr over S holds
( IT is properly-upper-bound iff the Family of IT is properly-upper-bound );

:: deftheorem Def10 defines properly-lower-bound CLOSURE1:def 10 :
for S being 1-sorted
for IT being MSClosureStr over S holds
( IT is properly-lower-bound iff the Family of IT is properly-lower-bound );

definition

let S be 1-sorted ;
let MS be many-sorted over S;

func MSFull MS -> MSClosureStr over S equals :: CLOSURE1:def 11
MSClosureStr(# the Sorts of MS,(bool the Sorts of MS) #);

correctness
coherence
MSClosureStr(# the Sorts of MS,(bool the Sorts of MS) #) is MSClosureStr over S;
;

end;

:: deftheorem defines MSFull CLOSURE1:def 11 :
for S being 1-sorted
for MS being many-sorted over S holds MSFull MS = MSClosureStr(# the Sorts of MS,(bool the Sorts of MS) #);

registration

let S be 1-sorted ;
let MS be many-sorted over S;

cluster MSFull MS -> strict additive absolutely-additive multiplicative absolutely-multiplicative properly-upper-bound properly-lower-bound ;

coherence
( MSFull MS is strict & MSFull MS is additive & MSFull MS is absolutely-additive & MSFull MS is multiplicative & MSFull MS is absolutely-multiplicative & MSFull MS is properly-upper-bound & MSFull MS is properly-lower-bound ) ;

end;

registration

let S be 1-sorted ;
let MS be non-empty many-sorted over S;

cluster MSFull MS -> non-empty ;

coherence
MSFull MS is non-empty ;

end;

registration

let S be 1-sorted ;

cluster non-empty strict additive absolutely-additive multiplicative absolutely-multiplicative properly-upper-bound properly-lower-bound for MSClosureStr over S;

existence
ex b₁ being MSClosureStr over S st
( b₁ is strict & b₁ is non-empty & b₁ is additive & b₁ is absolutely-additive & b₁ is multiplicative & b₁ is absolutely-multiplicative & b₁ is properly-upper-bound & b₁ is properly-lower-bound )

proof end;

end;

registration

let S be 1-sorted ;
let CS be additive MSClosureStr over S;

cluster the Family of CS -> additive ;

coherence
the Family of CS is additive by Def5;

end;

registration

let S be 1-sorted ;
let CS be absolutely-additive MSClosureStr over S;

cluster the Family of CS -> absolutely-additive ;

coherence
the Family of CS is absolutely-additive by Def6;

end;

registration

let S be 1-sorted ;
let CS be multiplicative MSClosureStr over S;

cluster the Family of CS -> multiplicative ;

coherence
the Family of CS is multiplicative by Def7;

end;

registration

let S be 1-sorted ;
let CS be absolutely-multiplicative MSClosureStr over S;

cluster the Family of CS -> absolutely-multiplicative ;

coherence
the Family of CS is absolutely-multiplicative by Def8;

end;

registration

let S be 1-sorted ;
let CS be properly-upper-bound MSClosureStr over S;

cluster the Family of CS -> properly-upper-bound ;

coherence
the Family of CS is properly-upper-bound by Def9;

end;

registration

let S be 1-sorted ;
let CS be properly-lower-bound MSClosureStr over S;

cluster the Family of CS -> properly-lower-bound ;

coherence
the Family of CS is properly-lower-bound by Def10;

end;

registration

let S be 1-sorted ;
let M be V8() ManySortedSet of the carrier of S;
let F be MSSubsetFamily of M;

cluster MSClosureStr(# M,F #) -> non-empty ;

coherence
MSClosureStr(# M,F #) is non-empty ;

end;

registration

let S be 1-sorted ;
let MS be many-sorted over S;
let F be additive MSSubsetFamily of the Sorts of MS;

cluster MSClosureStr(# the Sorts of MS,F #) -> additive ;

coherence
MSClosureStr(# the Sorts of MS,F #) is additive ;

end;

registration

let S be 1-sorted ;
let MS be many-sorted over S;
let F be absolutely-additive MSSubsetFamily of the Sorts of MS;

cluster MSClosureStr(# the Sorts of MS,F #) -> absolutely-additive ;

coherence
MSClosureStr(# the Sorts of MS,F #) is absolutely-additive ;

end;

registration

let S be 1-sorted ;
let MS be many-sorted over S;
let F be multiplicative MSSubsetFamily of the Sorts of MS;

cluster MSClosureStr(# the Sorts of MS,F #) -> multiplicative ;

coherence
MSClosureStr(# the Sorts of MS,F #) is multiplicative ;

end;

registration

let S be 1-sorted ;
let MS be many-sorted over S;
let F be absolutely-multiplicative MSSubsetFamily of the Sorts of MS;

cluster MSClosureStr(# the Sorts of MS,F #) -> absolutely-multiplicative ;

coherence
MSClosureStr(# the Sorts of MS,F #) is absolutely-multiplicative ;

end;

registration

let S be 1-sorted ;
let MS be many-sorted over S;
let F be properly-upper-bound MSSubsetFamily of the Sorts of MS;

cluster MSClosureStr(# the Sorts of MS,F #) -> properly-upper-bound ;

coherence
MSClosureStr(# the Sorts of MS,F #) is properly-upper-bound ;

end;

registration

let S be 1-sorted ;
let MS be many-sorted over S;
let F be properly-lower-bound MSSubsetFamily of the Sorts of MS;

cluster MSClosureStr(# the Sorts of MS,F #) -> properly-lower-bound ;

coherence
MSClosureStr(# the Sorts of MS,F #) is properly-lower-bound ;

end;

registration

let S be 1-sorted ;

cluster absolutely-additive -> additive for MSClosureStr over S;

coherence
for b₁ being MSClosureStr over S st b₁ is absolutely-additive holds
b₁ is additive ;

end;

registration

let S be 1-sorted ;

cluster absolutely-multiplicative -> multiplicative for MSClosureStr over S;

coherence
for b₁ being MSClosureStr over S st b₁ is absolutely-multiplicative holds
b₁ is multiplicative ;

end;

registration

let S be 1-sorted ;

cluster absolutely-multiplicative -> properly-upper-bound for MSClosureStr over S;

coherence
for b₁ being MSClosureStr over S st b₁ is absolutely-multiplicative holds
b₁ is properly-upper-bound ;

end;

registration

let S be 1-sorted ;

cluster absolutely-additive -> properly-lower-bound for MSClosureStr over S;

coherence
for b₁ being MSClosureStr over S st b₁ is absolutely-additive holds
b₁ is properly-lower-bound ;

end;

definition

let S be 1-sorted ;
mode MSClosureSystem of S is absolutely-multiplicative MSClosureStr over S;

end;

definition

let I be set ;
let M be ManySortedSet of I;
mode MSClosureOperator of M is reflexive monotonic idempotent MSSetOp of M;

end;

definition

let I be set ;
let M be ManySortedSet of I;
let F be ManySortedFunction of M,M;

func MSFixPoints F -> ManySortedSubset of M means :Def12: :: CLOSURE1:def 12
for x being set
for i being object st i in I holds
( x in it . i iff ex f being Function st
( f = F . i & x in dom f & f . x = x ) );

existence
ex b₁ being ManySortedSubset of M st
for x being set
for i being object st i in I holds
( x in b₁ . i iff ex f being Function st
( f = F . i & x in dom f & f . x = x ) )

proof end;

uniqueness
for b₁, b₂ being ManySortedSubset of M st ( for x being set
for i being object st i in I holds
( x in b₁ . i iff ex f being Function st
( f = F . i & x in dom f & f . x = x ) ) ) & ( for x being set
for i being object st i in I holds
( x in b₂ . i iff ex f being Function st
( f = F . i & x in dom f & f . x = x ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def12 defines MSFixPoints CLOSURE1:def 12 :
for I being set
for M being ManySortedSet of I
for F being ManySortedFunction of M,M
for b₄ being ManySortedSubset of M holds
( b₄ = MSFixPoints F iff for x being set
for i being object st i in I holds
( x in b₄ . i iff ex f being Function st
( f = F . i & x in dom f & f . x = x ) ) );

registration

let I be set ;
let M be V9() ManySortedSet of I;
let F be ManySortedFunction of M,M;

cluster MSFixPoints F -> V9() ;

coherence
MSFixPoints F is empty-yielding

proof end;

end;

theorem Th28: :: CLOSURE1:28

for I being set
for A, M being ManySortedSet of I
for F being ManySortedFunction of M,M holds
( ( A in M & F .. A = A ) iff A in MSFixPoints F )

proof end;

theorem :: CLOSURE1:29

for I being set
for A being ManySortedSet of I holds MSFixPoints (id A) = A

proof end;

theorem Th30: :: CLOSURE1:30

for S being 1-sorted
for A being ManySortedSet of the carrier of S
for J being reflexive monotonic MSSetOp of A
for D being MSSubsetFamily of A st D = MSFixPoints J holds
MSClosureStr(# A,D #) is MSClosureSystem of S

proof end;

theorem Th31: :: CLOSURE1:31

for I being set
for M being ManySortedSet of I
for D being properly-upper-bound MSSubsetFamily of M
for X being Element of bool M ex SF being V8() MSSubsetFamily of M st
for Y being ManySortedSet of I holds
( Y in SF iff ( Y in D & X c= Y ) )

proof end;

theorem Th32: :: CLOSURE1:32

for I being set
for M being ManySortedSet of I
for D being properly-upper-bound MSSubsetFamily of M
for X being Element of bool M
for SF being V8() MSSubsetFamily of M st ( for Y being ManySortedSet of I holds
( Y in SF iff ( Y in D & X c= Y ) ) ) holds
for i being set
for Di being non empty set st i in I & Di = D . i holds
SF . i = { z where z is Element of Di : X . i c= z }

proof end;

theorem Th33: :: CLOSURE1:33

for I being set
for M being ManySortedSet of I
for D being properly-upper-bound MSSubsetFamily of M ex J being MSSetOp of M st
for X being Element of bool M
for SF being V8() MSSubsetFamily of M st ( for Y being ManySortedSet of I holds
( Y in SF iff ( Y in D & X c= Y ) ) ) holds
J .. X = meet SF

proof end;

theorem Th34: :: CLOSURE1:34

for I being set
for M being ManySortedSet of I
for D being properly-upper-bound MSSubsetFamily of M
for A being Element of bool M
for J being MSSetOp of M st A in D & ( for X being Element of bool M
for SF being V8() MSSubsetFamily of M st ( for Y being ManySortedSet of I holds
( Y in SF iff ( Y in D & X c= Y ) ) ) holds
J .. X = meet SF ) holds
J .. A = A

proof end;

theorem :: CLOSURE1:35

for I being set
for M being ManySortedSet of I
for D being absolutely-multiplicative MSSubsetFamily of M
for A being Element of bool M
for J being MSSetOp of M st J .. A = A & ( for X being Element of bool M
for SF being V8() MSSubsetFamily of M st ( for Y being ManySortedSet of I holds
( Y in SF iff ( Y in D & X c= Y ) ) ) holds
J .. X = meet SF ) holds
A in D

proof end;

theorem Th36: :: CLOSURE1:36

for I being set
for M being ManySortedSet of I
for D being properly-upper-bound MSSubsetFamily of M
for J being MSSetOp of M st ( for X being Element of bool M
for SF being V8() MSSubsetFamily of M st ( for Y being ManySortedSet of I holds
( Y in SF iff ( Y in D & X c= Y ) ) ) holds
J .. X = meet SF ) holds
( J is reflexive & J is monotonic )

proof end;

theorem Th37: :: CLOSURE1:37

for I being set
for M being ManySortedSet of I
for D being absolutely-multiplicative MSSubsetFamily of M
for J being MSSetOp of M st ( for X being Element of bool M
for SF being V8() MSSubsetFamily of M st ( for Y being ManySortedSet of I holds
( Y in SF iff ( Y in D & X c= Y ) ) ) holds
J .. X = meet SF ) holds
J is idempotent

proof end;

theorem :: CLOSURE1:38

for S being 1-sorted
for D being MSClosureSystem of S
for J being MSSetOp of the Sorts of D st ( for X being Element of bool the Sorts of D
for SF being V8() MSSubsetFamily of the Sorts of D st ( for Y being ManySortedSet of the carrier of S holds
( Y in SF iff ( Y in the Family of D & X c= Y ) ) ) holds
J .. X = meet SF ) holds
J is MSClosureOperator of the Sorts of D by Th36, Th37;

definition

let S be 1-sorted ;
let A be ManySortedSet of the carrier of S;
let C be MSClosureOperator of A;

func ClOp->ClSys C -> MSClosureSystem of S means :Def13: :: CLOSURE1:def 13
ex D being MSSubsetFamily of A st
( D = MSFixPoints C & it = MSClosureStr(# A,D #) );

existence
ex b₁ being MSClosureSystem of S ex D being MSSubsetFamily of A st
( D = MSFixPoints C & b₁ = MSClosureStr(# A,D #) )

proof end;

uniqueness
for b₁, b₂ being MSClosureSystem of S st ex D being MSSubsetFamily of A st
( D = MSFixPoints C & b₁ = MSClosureStr(# A,D #) ) & ex D being MSSubsetFamily of A st
( D = MSFixPoints C & b₂ = MSClosureStr(# A,D #) ) holds
b₁ = b₂ ;

end;

:: deftheorem Def13 defines ClOp->ClSys CLOSURE1:def 13 :
for S being 1-sorted
for A being ManySortedSet of the carrier of S
for C being MSClosureOperator of A
for b₄ being MSClosureSystem of S holds
( b₄ = ClOp->ClSys C iff ex D being MSSubsetFamily of A st
( D = MSFixPoints C & b₄ = MSClosureStr(# A,D #) ) );

registration

let S be 1-sorted ;
let A be ManySortedSet of the carrier of S;
let C be MSClosureOperator of A;

cluster ClOp->ClSys C -> strict ;

coherence
ClOp->ClSys C is strict

proof end;

end;

registration

let S be 1-sorted ;
let A be V8() ManySortedSet of the carrier of S;
let C be MSClosureOperator of A;

cluster ClOp->ClSys C -> non-empty ;

coherence
ClOp->ClSys C is non-empty

proof end;

end;

definition

let S be 1-sorted ;
let D be MSClosureSystem of S;

func ClSys->ClOp D -> MSClosureOperator of the Sorts of D means :Def14: :: CLOSURE1:def 14
for X being Element of bool the Sorts of D
for SF being V8() MSSubsetFamily of the Sorts of D st ( for Y being ManySortedSet of the carrier of S holds
( Y in SF iff ( Y in the Family of D & X c= Y ) ) ) holds
it .. X = meet SF;

existence
ex b₁ being MSClosureOperator of the Sorts of D st
for X being Element of bool the Sorts of D
for SF being V8() MSSubsetFamily of the Sorts of D st ( for Y being ManySortedSet of the carrier of S holds
( Y in SF iff ( Y in the Family of D & X c= Y ) ) ) holds
b₁ .. X = meet SF

proof end;

uniqueness
for b₁, b₂ being MSClosureOperator of the Sorts of D st ( for X being Element of bool the Sorts of D
for SF being V8() MSSubsetFamily of the Sorts of D st ( for Y being ManySortedSet of the carrier of S holds
( Y in SF iff ( Y in the Family of D & X c= Y ) ) ) holds
b₁ .. X = meet SF ) & ( for X being Element of bool the Sorts of D
for SF being V8() MSSubsetFamily of the Sorts of D st ( for Y being ManySortedSet of the carrier of S holds
( Y in SF iff ( Y in the Family of D & X c= Y ) ) ) holds
b₂ .. X = meet SF ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def14 defines ClSys->ClOp CLOSURE1:def 14 :
for S being 1-sorted
for D being MSClosureSystem of S
for b₃ being MSClosureOperator of the Sorts of D holds
( b₃ = ClSys->ClOp D iff for X being Element of bool the Sorts of D
for SF being V8() MSSubsetFamily of the Sorts of D st ( for Y being ManySortedSet of the carrier of S holds
( Y in SF iff ( Y in the Family of D & X c= Y ) ) ) holds
b₃ .. X = meet SF );

theorem :: CLOSURE1:39

for S being 1-sorted
for A being ManySortedSet of the carrier of S
for J being MSClosureOperator of A holds ClSys->ClOp (ClOp->ClSys J) = J

proof end;

theorem :: CLOSURE1:40

for S being 1-sorted
for D being MSClosureSystem of S holds ClOp->ClSys (ClSys->ClOp D) = MSClosureStr(# the Sorts of D, the Family of D #)

proof end;