:: Finite Sets
:: by Agata Darmochwa\l
::
:: Copyright (c) 1990-2019 Association of Mizar Users

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 object holds {x} is finite
proof end;

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

registration
cluster empty -> finite for 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[ object ], F2( object ) -> object , F3( object ) -> object } :
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 object ;
cluster {x} -> finite ;
coherence
{x} is finite
by Lm1;
end;

registration
let x, y be object ;
cluster {x,y} -> finite ;
coherence
{x,y} is finite
proof end;
end;

registration
let x, y, z be object ;
cluster {x,y,z} -> finite ;
coherence
{x,y,z} is finite
proof end;
end;

registration
let x1, x2, x3, x4 be object ;
cluster {x1,x2,x3,x4} -> finite ;
coherence
{x1,x2,x3,x4} is finite
proof end;
end;

registration
let x1, x2, x3, x4, x5 be object ;
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 object ;
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 object ;
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 object ;
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 for 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
for A, B being set st A c= B & B is finite holds
A is finite ;

theorem :: FINSET_1:2
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_1:3
for A, B being set st A is finite holds
A /\ B is finite ;

theorem :: FINSET_1:4
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_1:5
for A being set
for f being Function st A is finite holds
f .: A is finite ;

theorem Th6: :: FINSET_1:6
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 ;
coherence
bool A is finite
proof end;
end;

theorem Th7: :: FINSET_1:7
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 Th8: :: FINSET_1:8
for f being Function st dom f is finite holds
rng f is finite
proof end;

theorem :: FINSET_1:9
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 for Element of bool X;
existence
ex b1 being Subset of X st
( b1 is finite & not b1 is empty )
proof end;
end;

:: from AMI_1
theorem Th10: :: FINSET_1:10
for f being Function holds
( dom f is finite iff f is finite )
proof end;

:: from ALI2
theorem :: FINSET_1:11
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;

:: from FIN_TOPO
theorem :: FINSET_1:12
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;

:: 2006.08.25, A.T.
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( object ) -> object = \$1 1 ;

theorem :: FINSET_1:13
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_1:14
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;

:: restored, 2007.07.22, A.T.
registration
existence
ex b1 being Function st
( b1 is finite & not b1 is empty )
proof end;
end;

registration
let R be finite Relation;
coherence
dom R is finite
proof end;
end;

:: from SCMFSA_4, 2007.07.22, A.T.
registration
let f be Function;
let g be finite Function;
cluster g * f -> finite ;
coherence
f * g is finite
proof end;
end;

:: from SF_MASTR, 2007.07.25, A.T.
registration
let A be finite set ;
let B be set ;
cluster Function-like V18(A,B) -> finite for Element of bool [:A,B:];
coherence
for b1 being Function of A,B holds b1 is finite
proof end;
end;

:: from GLIB_000, 2007.10.24, A.T.
registration
let x, y be object ;
cluster x .--> y -> finite ;
coherence
x .--> y is finite
;
end;

:: from FINSEQ_1, 2008.02.19, A.T.
registration
let R be finite Relation;
coherence
rng R is finite
proof end;
end;

:: from FINSEQ_1, 2008.05.06, A.T.
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;

:: from COMPTS_1, 2008.07.16, A.T
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 f be Function;
redefine attr f is finite-yielding means :Def4: :: FINSET_1:def 4
for i being object st i in dom f holds
f . i is finite ;
compatibility
( f is finite-yielding iff for i being object st i in dom f holds
f . i is finite )
proof end;
end;

:: deftheorem Def4 defines finite-yielding FINSET_1:def 4 :
for f being Function holds
( f is finite-yielding iff for i being object st i in dom f holds
f . i is finite );

:: from PRE_CIRC, 2009.03.04, A.T.
definition
let I be set ;
let IT be I -defined Function;
:: original: finite-yielding
redefine attr IT is finite-yielding means :: FINSET_1:def 5
for i being object st i in I holds
IT . i is finite ;
compatibility
( IT is finite-yielding iff for i being object st i in I holds
IT . i is finite )
proof end;
end;

:: deftheorem defines finite-yielding FINSET_1:def 5 :
for I being set
for IT being b1 -defined Function holds
( IT is finite-yielding iff for i being object st i in I holds
IT . i is finite );

:: new, 2009.08.26, A.T
theorem :: FINSET_1:15
for A, B being set st B is infinite holds
not B in [:A,B:]
proof end;

:: new, 2009.09.30, A.T.
registration
let I be set ;
let f be I -defined Function;
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 ;
existence
ex b1 being Function st
( b1 is finite & b1 is X -defined & b1 is Y -valued )
proof end;
end;

registration
let X, Y be non empty set ;
existence
ex b1 being Function st
( b1 is X -defined & b1 is Y -valued & not b1 is empty & b1 is finite )
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;
coherence
field R is finite
;
end;

registration
cluster trivial -> finite for set ;
coherence
for b1 being set st b1 is trivial holds
b1 is finite
proof end;
end;

registration
cluster infinite -> non trivial for set ;
coherence
for b1 being set st b1 is infinite holds
not b1 is trivial
;
end;

registration
let X be non trivial set ;
cluster non trivial finite for Element of bool X;
existence
ex b1 being Subset of X st
( b1 is finite & not b1 is trivial )
proof end;
end;

:: 2011.04.07, A.T,
registration
let x, y, a, b be object ;
cluster (x,y) --> (a,b) -> finite ;
coherence
(x,y) --> (a,b) is finite
;
end;

:: from MATROID0, 2011.07.25, A.T.
definition
let A be set ;
attr A is finite-membered means :Def6: :: FINSET_1:def 6
for B being set st B in A holds
B is finite ;
end;

:: deftheorem Def6 defines finite-membered FINSET_1:def 6 :
for A being set holds
( A is finite-membered iff for B being set st B in A holds
B is finite );

registration
coherence
for b1 being set st b1 is empty holds
b1 is finite-membered
;
end;

registration
let A be finite-membered set ;
cluster -> finite for Element of A;
coherence
for b1 being Element of A holds b1 is finite
proof end;
end;

registration
existence
ex b1 being set st
( not b1 is empty & b1 is finite & b1 is finite-membered )
proof end;
end;

:: from SIMPLEX0, 2011.07.25, A.T.
registration
let X be finite set ;
coherence by TARSKI:def 1;
coherence ;
let Y be finite set ;
cluster {X,Y} -> finite-membered ;
coherence
{X,Y} is finite-membered
by TARSKI:def 2;
end;

registration
let X be finite-membered set ;
cluster -> finite-membered for Element of bool X;
coherence
for b1 being Subset of X holds b1 is finite-membered
;
let Y be finite-membered set ;
coherence
proof end;
end;

registration
let X be finite finite-membered set ;
coherence
union X is finite
proof end;
end;

registration
existence
ex b1 being Function st
( not b1 is empty & b1 is finite-yielding )
proof end;
coherence
for b1 being Relation st b1 is empty holds
b1 is finite-yielding
;
end;

registration
let F be finite-yielding Function;
let x be object ;
cluster F . x -> finite ;
coherence
F . x is finite
proof end;
end;

registration
let F be finite-yielding Relation;
coherence by Def2;
end;