:: Definitions and Basic Properties of Boolean & Union of Many Sorted Sets
:: by Artur Korni{\l}owicz
::
:: Received April 27, 1995
:: Copyright (c) 1995-2012 Association of Mizar Users


begin

definition
let I be set ;
let A be ManySortedSet of I;
func bool A ->  ManySortedSet of I means :Def1: :: MBOOLEAN:def 1
 for i being set st i in I holds
it . i = bool (A . i);
existence
 ex b1 being ManySortedSet of I st
for i being set st i in I holds
b1 . i = bool (A . i)
proofend;
uniqueness
 for b1, b2 being ManySortedSet of I st ( for i being set st i in I holds
b1 . i = bool (A . i) ) & ( for i being set st i in I holds
b2 . i = bool (A . i) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def1 defines bool MBOOLEAN:def_1_:_
 for I being set
for A, b3 being ManySortedSet of I holds
( b3 = bool A iff for i being set st i in I holds
b3 . i = bool (A . i) );

registration
let I be set ;
let A be ManySortedSet of I;
cluster bool A -> V2() ;
coherence
 bool A is non-empty
proofend;
end;

Lm1:  for i, I, X being set
for M being ManySortedSet of I st i in I holds
dom (M +* (i .--> X)) = I
proofend;

Lm2:  for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(bool (A \/ B)) . i = bool ((A . i) \/ (B . i))
proofend;

Lm3:  for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(bool (A /\ B)) . i = bool ((A . i) /\ (B . i))
proofend;

Lm4:  for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(bool (A \ B)) . i = bool ((A . i) \ (B . i))
proofend;

Lm5:  for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(bool (A \+\ B)) . i = bool ((A . i) \+\ (B . i))
proofend;

theorem Th1: :: MBOOLEAN:1
for I being set
for X, Y being ManySortedSet of I holds
( X = bool Y iff for A being ManySortedSet of I holds
( A in X iff A c= Y ) )
proofend;

theorem :: MBOOLEAN:2
for I being set holds bool ([[0]] I) = I --> {{}}
proofend;

theorem :: MBOOLEAN:3
for I, x being set holds bool (I --> x) = I --> (bool x)
proofend;

theorem :: MBOOLEAN:4
for I, x being set holds bool (I --> {x}) = I --> {{},{x}}
proofend;

theorem :: MBOOLEAN:5
for I being set
for A being ManySortedSet of I holds [[0]] I c= A
proofend;

theorem :: MBOOLEAN:6
for I being set
for A, B being ManySortedSet of I st A c= B holds
bool A c= bool B
proofend;

theorem :: MBOOLEAN:7
for I being set
for A, B being ManySortedSet of I holds (bool A) \/ (bool B) c= bool (A \/ B)
proofend;

theorem :: MBOOLEAN:8
for I being set
for A, B being ManySortedSet of I st (bool A) \/ (bool B) = bool (A \/ B) holds
for i being set st i in I holds
A . i,B . i are_c=-comparable
proofend;

theorem :: MBOOLEAN:9
for I being set
for A, B being ManySortedSet of I holds bool (A /\ B) = (bool A) /\ (bool B)
proofend;

theorem :: MBOOLEAN:10
for I being set
for A, B being ManySortedSet of I holds bool (A \ B) c= (I --> {{}}) \/ ((bool A) \ (bool B))
proofend;

theorem :: MBOOLEAN:11
for I being set
for X, A, B being ManySortedSet of I holds
( X c= A \ B iff ( X c= A & X misses B ) )
proofend;

theorem :: MBOOLEAN:12
for I being set
for A, B being ManySortedSet of I holds (bool (A \ B)) \/ (bool (B \ A)) c= bool (A \+\ B)
proofend;

theorem :: MBOOLEAN:13
for I being set
for X, A, B being ManySortedSet of I holds
( X c= A \+\ B iff ( X c= A \/ B & X misses A /\ B ) )
proofend;

theorem :: MBOOLEAN:14
for I being set
for X, A, Y being ManySortedSet of I st ( X c= A or Y c= A ) holds
X /\ Y c= A
proofend;

theorem :: MBOOLEAN:15
for I being set
for X, A, Y being ManySortedSet of I st X c= A holds
X \ Y c= A
proofend;

theorem :: MBOOLEAN:16
for I being set
for X, A, Y being ManySortedSet of I st X c= A & Y c= A holds
X \+\ Y c= A
proofend;

theorem :: MBOOLEAN:17
for I being set
for X, Y being ManySortedSet of I holds [|X,Y|] c= bool (bool (X \/ Y))
proofend;

theorem :: MBOOLEAN:18
for I being set
for X, A being ManySortedSet of I holds
( X c= A iff X in bool A )
proofend;

theorem :: MBOOLEAN:19
for I being set
for A, B being ManySortedSet of I holds (Funcs) (A,B) c= bool [|A,B|]
proofend;

begin

definition
let I be set ;
let A be ManySortedSet of I;
func union A ->  ManySortedSet of I means :Def2: :: MBOOLEAN:def 2
 for i being set st i in I holds
it . i = union (A . i);
existence
 ex b1 being ManySortedSet of I st
for i being set st i in I holds
b1 . i = union (A . i)
proofend;
uniqueness
 for b1, b2 being ManySortedSet of I st ( for i being set st i in I holds
b1 . i = union (A . i) ) & ( for i being set st i in I holds
b2 . i = union (A . i) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def2 defines union MBOOLEAN:def_2_:_
 for I being set
for A, b3 being ManySortedSet of I holds
( b3 = union A iff for i being set st i in I holds
b3 . i = union (A . i) );

registration
let I be set ;
cluster union ([[0]] I) -> V3() ;
coherence
 union ([[0]] I) is empty-yielding
proofend;
end;

Lm6:  for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(union (A \/ B)) . i = union ((A . i) \/ (B . i))
proofend;

Lm7:  for I being set
for A, B being ManySortedSet of I
for i being set st i in I holds
(union (A /\ B)) . i = union ((A . i) /\ (B . i))
proofend;

theorem :: MBOOLEAN:20
for I being set
for A, X being ManySortedSet of I holds
( A in union X iff ex Y being ManySortedSet of I st
( A in Y & Y in X ) )
proofend;

theorem :: MBOOLEAN:21
for I being set holds union ([[0]] I) = [[0]] I
proofend;

theorem :: MBOOLEAN:22
for I, x being set holds union (I --> x) = I --> (union x)
proofend;

theorem :: MBOOLEAN:23
for I, x being set holds union (I --> {x}) = I --> x
proofend;

theorem :: MBOOLEAN:24
for I, x, y being set holds union (I --> {{x},{y}}) = I --> {x,y}
proofend;

theorem :: MBOOLEAN:25
for I being set
for X, A being ManySortedSet of I st X in A holds
X c= union A
proofend;

theorem :: MBOOLEAN:26
for I being set
for A, B being ManySortedSet of I st A c= B holds
union A c= union B
proofend;

theorem :: MBOOLEAN:27
for I being set
for A, B being ManySortedSet of I holds union (A \/ B) = (union A) \/ (union B)
proofend;

theorem :: MBOOLEAN:28
for I being set
for A, B being ManySortedSet of I holds union (A /\ B) c= (union A) /\ (union B)
proofend;

theorem :: MBOOLEAN:29
for I being set
for A being ManySortedSet of I holds union (bool A) = A
proofend;

theorem :: MBOOLEAN:30
for I being set
for A being ManySortedSet of I holds A c= bool (union A)
proofend;

theorem :: MBOOLEAN:31
for I being set
for Y, A, X being ManySortedSet of I st union Y c= A & X in Y holds
X c= A
proofend;

theorem :: MBOOLEAN:32
for I being set
for Z being ManySortedSet of I
for A being V2() ManySortedSet of I st ( for X being ManySortedSet of I st X in A holds
X c= Z ) holds
union A c= Z
proofend;

theorem :: MBOOLEAN:33
for I being set
for B being ManySortedSet of I
for A being V2() ManySortedSet of I st ( for X being ManySortedSet of I st X in A holds
X /\ B = [[0]] I ) holds
(union A) /\ B = [[0]] I
proofend;

theorem :: MBOOLEAN:34
for I being set
for A, B being ManySortedSet of I st A \/ B is V2() &amp; ( for X, Y being ManySortedSet of I st X <> Y &amp; X in A \/ B &amp; Y in A \/ B holds
X /\ Y = [[0]] I ) holds
union (A /\ B) = (union A) /\ (union B)
proofend;

theorem :: MBOOLEAN:35
for I being set
for A, X being ManySortedSet of I
for B being V2() ManySortedSet of I st X c= union (A \/ B) &amp; ( for Y being ManySortedSet of I st Y in B holds
Y /\ X = [[0]] I ) holds
X c= union A
proofend;

theorem :: MBOOLEAN:36
for I being set
for A being V2() V26() ManySortedSet of I st ( for X, Y being ManySortedSet of I st X in A &amp; Y in A &amp; not X c= Y holds
Y c= X ) holds
union A in A
proofend;