:: Algebraic Operation on Subsets of Many Sorted Sets
:: by Agnieszka Julia Marasik
::
:: Copyright (c) 1997-2019 Association of Mizar Users

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 ;

theorem Th3: :: CLOSURE3:3
for S being non empty non void ManySortedSign
for MA being strict non-empty MSAlgebra over 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;

theorem :: CLOSURE3:4
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;

Lm1: now :: thesis: for I being non empty set
for M being ManySortedSet of I holds support M = { x where x is Element of I : M . x <> {} }
let I be non empty set ; :: thesis: for M being ManySortedSet of I holds support M = { x where x is Element of I : M . x <> {} }
let M be ManySortedSet of I; :: thesis: support M = { x where x is Element of I : M . x <> {} }
set F = { x where x is Element of I : M . x <> {} } ;
thus support M = { x where x is Element of I : M . x <> {} } :: thesis: verum
proof
thus support M c= { x where x is Element of I : M . x <> {} } :: according to XBOOLE_0:def 10 :: thesis: { x where x is Element of I : M . x <> {} } c= support M
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in support M or x in { x where x is Element of I : M . x <> {} } )
A1: dom M = I by PARTFUN1:def 2;
A2: support M c= dom M by PRE_POLY:37;
assume A3: x in support M ; :: thesis: x in { x where x is Element of I : M . x <> {} }
then M . x <> {} by PRE_POLY:def 7;
hence x in { x where x is Element of I : M . x <> {} } by A1, A2, A3; :: thesis: verum
end;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { x where x is Element of I : M . x <> {} } or x in support M )
assume x in { x where x is Element of I : M . x <> {} } ; :: thesis:
then ex i being Element of I st
( x = i & M . i <> {} ) ;
hence x in support M by PRE_POLY:def 7; :: thesis: verum
end;
end;

definition
let I be non empty set ;
let M be ManySortedSet of I;
:: original: support
redefine func support M -> set equals :: CLOSURE3:def 1
{ x where x is Element of I : M . x <> {} } ;
compatibility
for b1 being set holds
( b1 = support M iff b1 = { x where x is Element of I : M . x <> {} } )
by Lm1;
end;

:: deftheorem defines support CLOSURE3:def 1 :
for I being non empty set
for M being ManySortedSet of I holds support M = { x where x is Element of I : M . x <> {} } ;

theorem :: CLOSURE3:5
for I being non empty set
for M being V8() ManySortedSet of I holds M = () +* (M | ())
proof end;

theorem :: CLOSURE3:6
for I being non empty set
for M1, M2 being V8() ManySortedSet of I st M1 | (support M1) = M2 | (support M2) holds
M1 = M2
proof end;

theorem :: CLOSURE3:7
canceled;

::\$CT
theorem Th7: :: CLOSURE3:8
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 V45() & support 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 :Def2: :: CLOSURE3:def 2
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 Def2 defines MSUnion CLOSURE3:def 2 :
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
proof end;
end;

theorem :: CLOSURE3:9
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 CLOSURE2:3;
end;

theorem :: CLOSURE3:10
for I being set
for M being ManySortedSet of I
for A, B being SubsetFamily of M holds MSUnion (A \/ B) = () (\/) ()
proof end;

theorem :: CLOSURE3:11
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 CLOSURE2:4;
end;

theorem :: CLOSURE3:12
for I being set
for M being ManySortedSet of I
for A, B being SubsetFamily of M holds MSUnion (A /\ B) c= () (/\) ()
proof end;

theorem :: CLOSURE3:13
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 = { () where X is SubsetFamily of M : X in AA } & A = union AA holds
MSUnion B = MSUnion A
proof end;

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 3
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 V45() & support a is finite & a c= x ) } & x = MSUnion A );
end;

:: deftheorem defines algebraic CLOSURE3:def 3 :
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 V45() & support a is finite & a c= x ) } & x = MSUnion A ) );

registration
let I be non empty set ;
let M be ManySortedSet of I;
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 4
ClSys->ClOp IT is algebraic ;
end;

:: deftheorem defines algebraic CLOSURE3:def 4 :
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 over S;
func SubAlgCl MA -> strict ClosureStr over 1-sorted(# the carrier of S #) means :Def5: :: CLOSURE3:def 5
( the Sorts of it = the Sorts of MA & the Family of it = SubSort MA );
existence
ex b1 being strict ClosureStr over 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 over 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 Def5 defines SubAlgCl CLOSURE3:def 5 :
for S being non empty non void ManySortedSign
for MA being non-empty MSAlgebra over S
for b3 being strict ClosureStr over 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 Th13: :: CLOSURE3:14
for S being non empty non void ManySortedSign
for MA being strict non-empty MSAlgebra over 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 over S;
coherence
proof end;
end;

registration
let S be non empty non void ManySortedSign ;
let MA be strict non-empty MSAlgebra over S;
coherence
proof end;
end;