:: Algebraic Operation on Subsets of Many Sorted Sets
:: by Agnieszka Julia Marasik
::
:: Received June 23, 1997
:: Copyright (c) 1997-2011 Association of Mizar Users

begin

registration

let S be non empty 1-sorted ;
cluster 1-sorted(# the carrier of S #) -> non empty ;
coherence
not 1-sorted(# the carrier of S #) is empty ;

end;

theorem Th1: :: CLOSURE3:1

for I being non empty set
for M, N being ManySortedSet of I holds M +* N = N

proof end;

theorem :: CLOSURE3:2

for I being set
for M, N being ManySortedSet of I
for F being SubsetFamily of M st N in F holds
meet |:F:| c=' N by CLOSURE2:22, MSSUBFAM:43;

theorem Th3: :: CLOSURE3:3

for S being non empty non void ManySortedSign
for MA being strict non-empty MSAlgebra of S
for F being SubsetFamily of the Sorts of MA st F c= SubSort MA holds
for B being MSSubset of MA st B = meet |:F:| holds
B is opers_closed

proof end;

begin

theorem :: CLOSURE3:4

canceled;

theorem :: CLOSURE3:5

for A, B, C being set st A is_coarser_than B & B is_coarser_than C holds
A is_coarser_than C

proof end;

definition

canceled;
canceled;
let I be non empty set ;
let M be ManySortedSet of I;
func supp M -> set means :Def3: :: CLOSURE3:def 3
it = { x where x is Element of I : M . x <> {} } ;
correctness
existence
ex b₁ being set st b₁ = { x where x is Element of I : M . x <> {} } ;
uniqueness
for b₁, b₂ being set st b₁ = { x where x is Element of I : M . x <> {} } & b₂ = { x where x is Element of I : M . x <> {} } holds
b₁ = b₂;
;

end;

:: deftheorem CLOSURE3:def 1 :
canceled;

:: deftheorem CLOSURE3:def 2 :
canceled;

:: deftheorem Def3 defines supp CLOSURE3:def 3 :
for I being non empty set
for M being ManySortedSet of I
for b₃ being set holds
( b₃ = supp M iff b₃ = { x where x is Element of I : M . x <> {} } );

theorem :: CLOSURE3:6

for I being non empty set
for M being V8() ManySortedSet of I holds M = ([[0]] I) +* (M | (supp M))

proof end;

theorem :: CLOSURE3:7

for I being non empty set
for M1, M2 being V8() ManySortedSet of I st M1 | (supp M1) = M2 | (supp M2) holds
M1 = M2

proof end;

theorem :: CLOSURE3:8

for I being non empty set
for M being ManySortedSet of I
for x being Element of I st not x in supp M holds
M . x = {}

proof end;

theorem Th9: :: CLOSURE3:9

for I being non empty set
for M being ManySortedSet of I
for x being Element of Bool M
for i being Element of I
for y being set st y in x . i holds
ex a being Element of Bool M st
( y in a . i & a is finite-yielding & supp a is finite & a c= x )

proof end;

definition

let I be set ;
let M be ManySortedSet of I;
let A be SubsetFamily of M;
func MSUnion A -> ManySortedSubset of M means :Def4: :: CLOSURE3:def 4
for i being set st i in I holds
it . i = union { (f . i) where f is Element of Bool M : f in A } ;
existence
ex b₁ being ManySortedSubset of M st
for i being set st i in I holds
b₁ . i = union { (f . i) where f is Element of Bool M : f in A }

proof end;

uniqueness
for b₁, b₂ being ManySortedSubset of M st ( for i being set st i in I holds
b₁ . i = union { (f . i) where f is Element of Bool M : f in A } ) & ( for i being set st i in I holds
b₂ . i = union { (f . i) where f is Element of Bool M : f in A } ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines MSUnion CLOSURE3:def 4 :
for I being set
for M being ManySortedSet of I
for A being SubsetFamily of M
for b₄ being ManySortedSubset of M holds
( b₄ = MSUnion A iff for i being set st i in I holds
b₄ . i = union { (f . i) where f is Element of Bool M : f in A } );

registration

let I be set ;
let M be ManySortedSet of I;
let A be empty SubsetFamily of M;
cluster MSUnion A -> V9() ;
coherence
MSUnion A is empty-yielding

proof end;

end;

theorem :: CLOSURE3:10

for I being set
for M being ManySortedSet of I
for A being SubsetFamily of M holds MSUnion A = union |:A:|

proof end;

definition

let I be set ;
let M be ManySortedSet of I;
let A, B be SubsetFamily of M;
:: original: \/
redefine func A \/ B -> SubsetFamily of M;
correctness
coherence
A \/ B is SubsetFamily of M;
by CLOSURE2:4;

end;

theorem :: CLOSURE3:11

for I being set
for M being ManySortedSet of I
for A, B being SubsetFamily of M holds MSUnion (A \/ B) = (MSUnion A) \/ (MSUnion B)

proof end;

theorem :: CLOSURE3:12

for I being set
for M being ManySortedSet of I
for A, B being SubsetFamily of M st A c= B holds
MSUnion A c= MSUnion B

proof end;

definition

let I be set ;
let M be ManySortedSet of I;
let A, B be SubsetFamily of M;
:: original: /\
redefine func A /\ B -> SubsetFamily of M;
correctness
coherence
A /\ B is SubsetFamily of M;
by CLOSURE2:5;

end;

theorem :: CLOSURE3:13

for I being set
for M being ManySortedSet of I
for A, B being SubsetFamily of M holds MSUnion (A /\ B) c= (MSUnion A) /\ (MSUnion B)

proof end;

theorem :: CLOSURE3:14

for I being set
for M being ManySortedSet of I
for AA being set st ( for x being set st x in AA holds
x is SubsetFamily of M ) holds
for A, B being SubsetFamily of M st B = { (MSUnion X) where X is SubsetFamily of M : X in AA } & A = union AA holds
MSUnion B = MSUnion A

proof end;

begin

definition

let I be non empty set ;
let M be ManySortedSet of I;
let S be SetOp of M;
attr S is algebraic means :: CLOSURE3:def 5
for x being Element of Bool M st x = S . x holds
ex A being SubsetFamily of M st
( A = { (S . a) where a is Element of Bool M : ( a is finite-yielding & supp a is finite & a c= x ) } & x = MSUnion A );

end;

:: deftheorem defines algebraic CLOSURE3:def 5 :
for I being non empty set
for M being ManySortedSet of I
for S being SetOp of M holds
( S is algebraic iff for x being Element of Bool M st x = S . x holds
ex A being SubsetFamily of M st
( A = { (S . a) where a is Element of Bool M : ( a is finite-yielding & supp a is finite & a c= x ) } & x = MSUnion A ) );

registration

let I be non empty set ;
let M be ManySortedSet of I;
cluster non empty Relation-like Bool M -defined Bool M -valued Function-like V17( Bool M) quasi_total reflexive monotonic idempotent algebraic Element of bool [:(Bool M),(Bool M):];
existence
ex b₁ being SetOp of M st
( b₁ is algebraic & b₁ is reflexive & b₁ is monotonic & b₁ is idempotent )

proof end;

end;

definition

let S be non empty 1-sorted ;
let IT be ClosureSystem of S;
attr IT is algebraic means :: CLOSURE3:def 6
ClSys->ClOp IT is algebraic ;

end;

:: deftheorem defines algebraic CLOSURE3:def 6 :
for S being non empty 1-sorted
for IT being ClosureSystem of S holds
( IT is algebraic iff ClSys->ClOp IT is algebraic );

definition

let S be non empty non void ManySortedSign ;
let MA be non-empty MSAlgebra of S;
func SubAlgCl MA -> strict ClosureStr of 1-sorted(# the carrier of S #) means :Def7: :: CLOSURE3:def 7
( the Sorts of it = the Sorts of MA & the Family of it = SubSort MA );
existence
ex b₁ being strict ClosureStr of 1-sorted(# the carrier of S #) st
( the Sorts of b₁ = the Sorts of MA & the Family of b₁ = SubSort MA )

proof end;

uniqueness
for b₁, b₂ being strict ClosureStr of 1-sorted(# the carrier of S #) st the Sorts of b₁ = the Sorts of MA & the Family of b₁ = SubSort MA & the Sorts of b₂ = the Sorts of MA & the Family of b₂ = SubSort MA holds
b₁ = b₂ ;

end;

:: deftheorem Def7 defines SubAlgCl CLOSURE3:def 7 :
for S being non empty non void ManySortedSign
for MA being non-empty MSAlgebra of S
for b₃ being strict ClosureStr of 1-sorted(# the carrier of S #) holds
( b₃ = SubAlgCl MA iff ( the Sorts of b₃ = the Sorts of MA & the Family of b₃ = SubSort MA ) );

theorem :: CLOSURE3:15

canceled;

theorem Th16: :: CLOSURE3:16

for S being non empty non void ManySortedSign
for MA being strict non-empty MSAlgebra of S holds SubSort MA is absolutely-multiplicative SubsetFamily of the Sorts of MA

proof end;

registration

let S be non empty non void ManySortedSign ;
let MA be strict non-empty MSAlgebra of S;
cluster SubAlgCl MA -> strict absolutely-multiplicative ;
coherence
SubAlgCl MA is absolutely-multiplicative

proof end;

end;

registration

let S be non empty non void ManySortedSign ;
let MA be strict non-empty MSAlgebra of S;
cluster SubAlgCl MA -> strict algebraic ;
coherence
SubAlgCl MA is algebraic

proof end;

end;