:: Finite Sets
:: by Agata Darmochwa\l
::
:: Received April 6, 1989
:: Copyright (c) 1990-2011 Association of Mizar Users

begin

definition

let IT be set ;
attr IT is finite means :Def1: :: FINSET_1:def 1
ex p being Function st
( rng p = IT & dom p in omega );

end;

:: deftheorem Def1 defines finite FINSET_1:def 1 :
for IT being set holds
( IT is finite iff ex p being Function st
( rng p = IT & dom p in omega ) );

notation

let IT be set ;
antonym infinite IT for finite ;

end;

Lm1: for x being set holds {x} is finite

proof end;

registration

cluster non empty finite set ;
existence
ex b₁ being set st
( not b₁ is empty & b₁ is finite )

proof end;

end;

registration

cluster empty -> finite set ;
coherence
for b₁ being set st b₁ is empty holds
b₁ is finite

proof end;

end;

scheme :: FINSET_1:sch 1
OLambdaC{ F₁() -> set , P₁[ set ], F₂( set ) -> set , F₃( set ) -> set } :

ex f being Function st
( dom f = F₁() & ( for x being Ordinal st x in F₁() holds
( ( P₁[x] implies f . x = F₂(x) ) & ( P₁[x] implies f . x = F₃(x) ) ) ) )

proof end;

Lm2: for A, B being set st A is finite & B is finite holds
A \/ B is finite

proof end;

registration

let x be set ;
cluster {x} -> finite ;
coherence
{x} is finite by Lm1;

end;

registration

let x, y be set ;
cluster {x,y} -> finite ;
coherence
{x,y} is finite

proof end;

end;

registration

let x, y, z be set ;
cluster {x,y,z} -> finite ;
coherence
{x,y,z} is finite

proof end;

end;

registration

let x1, x2, x3, x4 be set ;
cluster {x1,x2,x3,x4} -> finite ;
coherence
{x1,x2,x3,x4} is finite

proof end;

end;

registration

let x1, x2, x3, x4, x5 be set ;
cluster {x1,x2,x3,x4,x5} -> finite ;
coherence
{x1,x2,x3,x4,x5} is finite

proof end;

end;

registration

let x1, x2, x3, x4, x5, x6 be set ;
cluster {x1,x2,x3,x4,x5,x6} -> finite ;
coherence
{x1,x2,x3,x4,x5,x6} is finite

proof end;

end;

registration

let x1, x2, x3, x4, x5, x6, x7 be set ;
cluster {x1,x2,x3,x4,x5,x6,x7} -> finite ;
coherence
{x1,x2,x3,x4,x5,x6,x7} is finite

proof end;

end;

registration

let x1, x2, x3, x4, x5, x6, x7, x8 be set ;
cluster {x1,x2,x3,x4,x5,x6,x7,x8} -> finite ;
coherence
{x1,x2,x3,x4,x5,x6,x7,x8} is finite

proof end;

end;

registration

let B be finite set ;
cluster -> finite Element of bool B;
coherence
for b₁ being Subset of B holds b₁ is finite

proof end;

end;

registration

let X, Y be finite set ;
cluster X \/ Y -> finite ;
coherence
X \/ Y is finite by Lm2;

end;

theorem :: FINSET_1:1

canceled;

theorem :: FINSET_1:2

canceled;

theorem :: FINSET_1:3

canceled;

theorem :: FINSET_1:4

canceled;

theorem :: FINSET_1:5

canceled;

theorem :: FINSET_1:6

canceled;

theorem :: FINSET_1:7

canceled;

theorem :: FINSET_1:8

canceled;

theorem :: FINSET_1:9

canceled;

theorem :: FINSET_1:10

canceled;

theorem :: FINSET_1:11

canceled;

theorem :: FINSET_1:12

canceled;

theorem :: FINSET_1:13

for A, B being set st A c= B & B is finite holds
A is finite ;

theorem :: FINSET_1:14

for A, B being set st A is finite & B is finite holds
A \/ B is finite ;

registration

let A be set ;
let B be finite set ;
cluster A /\ B -> finite ;
coherence
A /\ B is finite

proof end;

end;

registration

let A be finite set ;
let B be set ;
cluster A /\ B -> finite ;
coherence
A /\ B is finite ;
cluster A \ B -> finite ;
coherence
A \ B is finite ;

end;

theorem :: FINSET_1:15

for A, B being set st A is finite holds
A /\ B is finite ;

theorem :: FINSET_1:16

for A, B being set st A is finite holds
A \ B is finite ;

registration

let f be Function;
let A be finite set ;
cluster f .: A -> finite ;
coherence
f .: A is finite

proof end;

end;

theorem :: FINSET_1:17

for A being set
for f being Function st A is finite holds
f .: A is finite ;

theorem Th18: :: FINSET_1:18

for A being set st A is finite holds
for X being Subset-Family of A st X <> {} holds
ex x being set st
( x in X & ( for B being set st B in X & x c= B holds
B = x ) )

proof end;

scheme :: FINSET_1:sch 2
Finite{ F₁() -> set , P₁[ set ] } :

P₁[F₁()]

provided

A1: F₁() is finite and
A2: P₁[ {} ] and
A3: for x, B being set st x in F₁() & B c= F₁() & P₁[B] holds
P₁[B \/ {x}]

proof end;

Lm3: for A being set st A is finite & ( for X being set st X in A holds
X is finite ) holds
union A is finite

proof end;

registration

let A, B be finite set ;
cluster [:A,B:] -> finite ;
coherence
[:A,B:] is finite

proof end;

end;

registration

let A, B, C be finite set ;
cluster [:A,B,C:] -> finite ;
coherence
[:A,B,C:] is finite

proof end;

end;

registration

let A, B, C, D be finite set ;
cluster [:A,B,C,D:] -> finite ;
coherence
[:A,B,C,D:] is finite

proof end;

end;

registration

let A be finite set ;
cluster bool A -> finite ;
coherence
bool A is finite

proof end;

end;

theorem :: FINSET_1:19

canceled;

theorem :: FINSET_1:20

canceled;

theorem :: FINSET_1:21

canceled;

theorem :: FINSET_1:22

canceled;

theorem :: FINSET_1:23

canceled;

theorem :: FINSET_1:24

canceled;

theorem :: FINSET_1:25

for A being set holds
( ( A is finite & ( for X being set st X in A holds
X is finite ) ) iff union A is finite )

proof end;

theorem Th26: :: FINSET_1:26

for f being Function st dom f is finite holds
rng f is finite

proof end;

theorem :: FINSET_1:27

for Y being set
for f being Function st Y c= rng f & f " Y is finite holds
Y is finite

proof end;

registration

let X, Y be finite set ;
cluster X \+\ Y -> finite ;
coherence
X \+\ Y is finite ;

end;

registration

let X be non empty set ;
cluster non empty finite Element of bool X;
existence
ex b₁ being Subset of X st
( b₁ is finite & not b₁ is empty )

proof end;

end;

begin

theorem :: FINSET_1:28

canceled;

theorem Th29: :: FINSET_1:29

for f being Function holds
( dom f is finite iff f is finite )

proof end;

theorem :: FINSET_1:30

for F being set st F is finite & F <> {} & F is c=-linear holds
ex m being set st
( m in F & ( for C being set st C in F holds
m c= C ) )

proof end;

theorem :: FINSET_1:31

for F being set st F is finite & F <> {} & F is c=-linear holds
ex m being set st
( m in F & ( for C being set st C in F holds
C c= m ) )

proof end;

definition

let R be Relation;
attr R is finite-yielding means :Def2: :: FINSET_1:def 2
for x being set st x in rng R holds
x is finite ;

end;

:: deftheorem Def2 defines finite-yielding FINSET_1:def 2 :
for R being Relation holds
( R is finite-yielding iff for x being set st x in rng R holds
x is finite );

deffunc H₁( set ) -> set = $1 `1 ;

theorem :: FINSET_1:32

for X, Y, Z being set st X is finite & X c= [:Y,Z:] holds
ex A, B being set st
( A is finite & A c= Y & B is finite & B c= Z & X c= [:A,B:] )

proof end;

theorem :: FINSET_1:33

for X, Y, Z being set st X is finite & X c= [:Y,Z:] holds
ex A being set st
( A is finite & A c= Y & X c= [:A,Z:] )

proof end;

registration

cluster non empty Relation-like Function-like finite set ;
existence
ex b₁ being Function st
( b₁ is finite & not b₁ is empty )

proof end;

end;

registration

let R be finite Relation;
cluster proj1 R -> finite ;
coherence
dom R is finite

proof end;

end;

registration

let f be Function;
let g be finite Function;
cluster g * f -> finite ;
coherence
f * g is finite

proof end;

end;

registration

let A be finite set ;
let B be set ;
cluster Function-like V15(A,B) -> finite Element of bool [:A,B:];
coherence
for b₁ being Function of A,B holds b₁ is finite

proof end;

end;

registration

let x, y be set ;
cluster x .--> y -> finite ;
coherence
x .--> y is finite ;

end;

registration

let R be finite Relation;
cluster proj2 R -> finite ;
coherence
rng R is finite

proof end;

end;

registration

let f be finite Function;
let x be set ;
cluster f " x -> finite ;
coherence
f " x is finite

proof end;

end;

registration

let f, g be finite Function;
cluster f +* g -> finite ;
coherence
f +* g is finite

proof end;

end;

definition

let F be set ;
attr F is centered means :: FINSET_1:def 3
( F <> {} & ( for G being set st G <> {} & G c= F & G is finite holds
meet G <> {} ) );

end;

:: deftheorem defines centered FINSET_1:def 3 :
for F being set holds
( F is centered iff ( F <> {} & ( for G being set st G <> {} & G c= F & G is finite holds
meet G <> {} ) ) );

definition

let I be set ;
let IT be I -defined Function;
:: original: finite-yielding
redefine attr IT is finite-yielding means :: FINSET_1:def 4
for i being set st i in I holds
IT . i is finite ;
compatibility
( IT is finite-yielding iff for i being set st i in I holds
IT . i is finite )

proof end;

end;

:: deftheorem defines finite-yielding FINSET_1:def 4 :
for I being set
for IT being b₁ -defined Function holds
( IT is finite-yielding iff for i being set st i in I holds
IT . i is finite );

theorem :: FINSET_1:34

for B, A being set st not B is finite holds
not B in [:A,B:]

proof end;

registration

let I be set ;
let f be I -defined Function;
cluster Relation-like I -defined Function-like f -compatible finite set ;
existence
ex b₁ being Function st
( b₁ is finite & b₁ is I -defined & b₁ is f -compatible )

proof end;

end;

registration

let X, Y be set ;
cluster Relation-like X -defined Y -valued Function-like finite set ;
existence
ex b₁ being Function st
( b₁ is finite & b₁ is X -defined & b₁ is Y -valued )

proof end;

end;

registration

let A be set ;
let F be finite Relation;
cluster A | F -> finite ;
coherence
A | F is finite

proof end;

end;

registration

let A be set ;
let F be finite Relation;
cluster F | A -> finite ;
coherence
F | A is finite

proof end;

end;

registration

let A be finite set ;
let F be Function;
cluster F | A -> finite ;
coherence
F | A is finite

proof end;

end;

registration

let R be finite Relation;
cluster field R -> finite ;
coherence
field R is finite ;

end;