SETWISEO: Semilattice Operations on Finite Subsets

:: Semilattice Operations on Finite Subsets
:: by Andrzej Trybulec
::
:: Received September 18, 1989
:: Copyright (c) 1990-2021 Association of Mizar Users

theorem Th1: :: SETWISEO:1

for x, y, z being set holds {x} c= {x,y,z}

proof end;

theorem Th2: :: SETWISEO:2

for x, y, z being set holds {x,y} c= {x,y,z}

proof end;

theorem :: SETWISEO:3

canceled;

theorem :: SETWISEO:4

canceled;

theorem :: SETWISEO:5

canceled;

::$CT 3

theorem Th3: :: SETWISEO:6

for X, Y being set
for f being Function holds f .: (Y \ (f " X)) = (f .: Y) \ X

proof end;

theorem Th4: :: SETWISEO:7

for X, Y being non empty set
for f being Function of X,Y
for x being Element of X holds x in f " {(f . x)}

proof end;

theorem Th5: :: SETWISEO:8

for X, Y being non empty set
for f being Function of X,Y
for x being Element of X holds Im (f,x) = {(f . x)}

proof end;

:: Theorems related to Fin ...

theorem Th6: :: SETWISEO:9

for X being non empty set
for B being Element of Fin X
for x being set st x in B holds
x is Element of X

proof end;

theorem :: SETWISEO:10

for X, Y being non empty set
for A being Element of Fin X
for B being set
for f being Function of X,Y st ( for x being Element of X st x in A holds
f . x in B ) holds
f .: A c= B

proof end;

theorem Th8: :: SETWISEO:11

for X being set
for B being Element of Fin X
for A being set st A c= B holds
A is Element of Fin X

proof end;

Lm1: for X, Y being non empty set
for f being Function of X,Y
for A being Element of Fin X holds f .: A is Element of Fin Y
by FINSUB_1:def 5;

theorem Th9: :: SETWISEO:12

for X being non empty set
for B being Element of Fin X st B <> {} holds
ex x being Element of X st x in B

proof end;

theorem Th10: :: SETWISEO:13

for X, Y being non empty set
for f being Function of X,Y
for A being Element of Fin X st f .: A = {} holds
A = {}

proof end;

registration

let X be set ;

cluster empty finite for Element of Fin X;

existence

proof end;

end;

definition

let X be set ;

func {}. X -> empty Element of Fin X equals :: SETWISEO:def 1
{} ;

coherence by FINSUB_1:7;

end;

:: deftheorem defines {}. SETWISEO:def 1 :

scheme :: SETWISEO:sch 1
FinSubFuncEx{ F₁() -> non empty set , F₂() -> Element of Fin F₁(), P₁[ set , set ] } :

ex f being Function of F₁(),(Fin F₁()) st
for b, a being Element of F₁() holds
( a in f . b iff ( a in F₂() & P₁[a,b] ) )

proof end;

:: theorems related to BinOp of ...

definition

let X be non empty set ;
let F be BinOp of X;

attr F is having_a_unity means :: SETWISEO:def 2
ex x being Element of X st x is_a_unity_wrt F;

end;

:: deftheorem defines having_a_unity SETWISEO:def 2 :

theorem Th11: :: SETWISEO:14

for X being non empty set
for F being BinOp of X holds
( F is having_a_unity iff the_unity_wrt F is_a_unity_wrt F ) by BINOP_1:def 8;

theorem Th12: :: SETWISEO:15

for X being non empty set
for F being BinOp of X st F is having_a_unity holds
for x being Element of X holds
( F . ((the_unity_wrt F),x) = x & F . (x,(the_unity_wrt F)) = x )

proof end;

:: Main part

registration

let X be non empty set ;

cluster non empty finite for Element of Fin X;

existence

proof end;

end;

notation

let X be non empty set ;
let x be Element of X;
synonym {.x.} for {X};
let y be Element of X;
synonym {.x,y.} for {X,x};
let z be Element of X;
synonym {.x,y,z.} for {X,x,y};

end;

definition

let X be non empty set ;
let x be Element of X;
:: original: {.
redefine func {.x.} -> Element of Fin X;
coherence

proof end;

let y be Element of X;
:: original: {.
redefine func {.x,y.} -> Element of Fin X;
coherence

proof end;

let z be Element of X;
:: original: {.
redefine func {.x,y,z.} -> Element of Fin X;
coherence

proof end;

end;

definition

let X be set ;
let A, B be Element of Fin X;
:: original: \/
redefine func A \/ B -> Element of Fin X;
coherence by FINSUB_1:1;

end;

definition

let X be set ;
let A, B be Element of Fin X;
:: original: \
redefine func A \ B -> Element of Fin X;
coherence by FINSUB_1:1;

end;

scheme :: SETWISEO:sch 2
FinSubInd1{ F₁() -> non empty set , P₁[ set ] } :

for B being Element of Fin F₁() holds P₁[B]

provided

A1: P₁[ {}. F₁()] and
A2: for B9 being Element of Fin F₁()
for b being Element of F₁() st P₁[B9] & not b in B9 holds
P₁[B9 \/ {b}]

proof end;

scheme :: SETWISEO:sch 3
FinSubInd2{ F₁() -> non empty set , P₁[ Element of Fin F₁()] } :

for B being non empty Element of Fin F₁() holds P₁[B]

provided

A1: for x being Element of F₁() holds P₁[{.x.}] and
A2: for B1, B2 being non empty Element of Fin F₁() st P₁[B1] & P₁[B2] holds
P₁[B1 \/ B2]

proof end;

scheme :: SETWISEO:sch 4
FinSubInd3{ F₁() -> non empty set , P₁[ set ] } :

for B being Element of Fin F₁() holds P₁[B]

provided

A1: P₁[ {}. F₁()] and
A2: for B9 being Element of Fin F₁()
for b being Element of F₁() st P₁[B9] holds
P₁[B9 \/ {b}]

proof end;

definition

let X, Y be non empty set ;
let F be BinOp of Y;
let B be Element of Fin X;
let f be Function of X,Y;
assume that
A1: ( B <> {} or F is having_a_unity ) and
A2: F is commutative and
A3: F is associative ;

func F $$ (B,f) -> Element of Y means :Def3: :: SETWISEO:def 3
ex G being Function of (Fin X),Y st
( it = G . B & ( for e being Element of Y st e is_a_unity_wrt F holds
G . {} = e ) & ( for x being Element of X holds G . {x} = f . x ) & ( for B9 being Element of Fin X st B9 c= B & B9 <> {} holds
for x being Element of X st x in B \ B9 holds
G . (B9 \/ {x}) = F . ((G . B9),(f . x)) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def3 defines $$ SETWISEO:def 3 :

theorem Th13: :: SETWISEO:16

for X, Y being non empty set
for F being BinOp of Y
for B being Element of Fin X
for f being Function of X,Y st ( B <> {} or F is having_a_unity ) & F is idempotent & F is commutative & F is associative holds
for IT being Element of Y holds
( IT = F $$ (B,f) iff ex G being Function of (Fin X),Y st
( IT = G . B & ( for e being Element of Y st e is_a_unity_wrt F holds
G . {} = e ) & ( for x being Element of X holds G . {x} = f . x ) & ( for B9 being Element of Fin X st B9 c= B & B9 <> {} holds
for x being Element of X st x in B holds
G . (B9 \/ {x}) = F . ((G . B9),(f . x)) ) ) )

proof end;

theorem Th14: :: SETWISEO:17

for X, Y being non empty set
for F being BinOp of Y
for f being Function of X,Y st F is commutative & F is associative holds
for b being Element of X holds F $$ ({.b.},f) = f . b

proof end;

theorem Th15: :: SETWISEO:18

for X, Y being non empty set
for F being BinOp of Y
for f being Function of X,Y st F is idempotent & F is commutative & F is associative holds
for a, b being Element of X holds F $$ ({.a,b.},f) = F . ((f . a),(f . b))

proof end;

theorem Th16: :: SETWISEO:19

for X, Y being non empty set
for F being BinOp of Y
for f being Function of X,Y st F is idempotent & F is commutative & F is associative holds
for a, b, c being Element of X holds F $$ ({.a,b,c.},f) = F . ((F . ((f . a),(f . b))),(f . c))

proof end;

:: If B is non empty

theorem Th17: :: SETWISEO:20

for X, Y being non empty set
for F being BinOp of Y
for B being Element of Fin X
for f being Function of X,Y st F is idempotent & F is commutative & F is associative & B <> {} holds
for x being Element of X holds F $$ ((B \/ {.x.}),f) = F . ((F $$ (B,f)),(f . x))

proof end;

theorem Th18: :: SETWISEO:21

for X, Y being non empty set
for F being BinOp of Y
for f being Function of X,Y st F is idempotent & F is commutative & F is associative holds
for B1, B2 being Element of Fin X st B1 <> {} & B2 <> {} holds
F $$ ((B1 \/ B2),f) = F . ((F $$ (B1,f)),(F $$ (B2,f)))

proof end;

theorem :: SETWISEO:22

for X, Y being non empty set
for F being BinOp of Y
for B being Element of Fin X
for f being Function of X,Y st F is commutative & F is associative & F is idempotent holds
for x being Element of X st x in B holds
F . ((f . x),(F $$ (B,f))) = F $$ (B,f)

proof end;

theorem :: SETWISEO:23

for X, Y being non empty set
for F being BinOp of Y
for f being Function of X,Y st F is commutative & F is associative & F is idempotent holds
for B, C being Element of Fin X st B <> {} & B c= C holds
F . ((F $$ (B,f)),(F $$ (C,f))) = F $$ (C,f)

proof end;

theorem Th21: :: SETWISEO:24

for X, Y being non empty set
for F being BinOp of Y
for B being Element of Fin X
for f being Function of X,Y st B <> {} & F is commutative & F is associative & F is idempotent holds
for a being Element of Y st ( for b being Element of X st b in B holds
f . b = a ) holds
F $$ (B,f) = a

proof end;

theorem Th22: :: SETWISEO:25

for X, Y being non empty set
for F being BinOp of Y
for B being Element of Fin X
for f being Function of X,Y st F is commutative & F is associative & F is idempotent holds
for a being Element of Y st f .: B = {a} holds
F $$ (B,f) = a

proof end;

theorem Th23: :: SETWISEO:26

for X, Y being non empty set
for F being BinOp of Y st F is commutative & F is associative & F is idempotent holds
for f, g being Function of X,Y
for A, B being Element of Fin X st A <> {} & f .: A = g .: B holds
F $$ (A,f) = F $$ (B,g)

proof end;

theorem :: SETWISEO:27

for X, Y being non empty set
for F, G being BinOp of Y st F is idempotent & F is commutative & F is associative & G is_distributive_wrt F holds
for B being Element of Fin X st B <> {} holds
for f being Function of X,Y
for a being Element of Y holds G . (a,(F $$ (B,f))) = F $$ (B,(G [;] (a,f)))

proof end;

theorem :: SETWISEO:28

proof end;

definition

let X, Y be non empty set ;
let f be Function of X,Y;
let A be Element of Fin X;
:: original: .:
redefine func f .: A -> Element of Fin Y;
coherence by Lm1;

end;

theorem Th26: :: SETWISEO:29

for A, X, Y being non empty set
for F being BinOp of A st F is idempotent & F is commutative & F is associative holds
for B being Element of Fin X st B <> {} holds
for f being Function of X,Y
for g being Function of Y,A holds F $$ ((f .: B),g) = F $$ (B,(g * f))

proof end;

theorem Th27: :: SETWISEO:30

for X, Y being non empty set
for F being BinOp of Y st F is commutative & F is associative & F is idempotent holds
for Z being non empty set
for G being BinOp of Z st G is commutative & G is associative & G is idempotent holds
for f being Function of X,Y
for g being Function of Y,Z st ( for x, y being Element of Y holds g . (F . (x,y)) = G . ((g . x),(g . y)) ) holds
for B being Element of Fin X st B <> {} holds
g . (F $$ (B,f)) = G $$ (B,(g * f))

proof end;

:: If F has a unity

theorem Th28: :: SETWISEO:31

for X, Y being non empty set
for F being BinOp of Y st F is commutative & F is associative & F is having_a_unity holds
for f being Function of X,Y holds F $$ (({}. X),f) = the_unity_wrt F

proof end;

theorem Th29: :: SETWISEO:32

for X, Y being non empty set
for F being BinOp of Y
for B being Element of Fin X
for f being Function of X,Y st F is idempotent & F is commutative & F is associative & F is having_a_unity holds
for x being Element of X holds F $$ ((B \/ {.x.}),f) = F . ((F $$ (B,f)),(f . x))

proof end;

theorem :: SETWISEO:33

for X, Y being non empty set
for F being BinOp of Y
for f being Function of X,Y st F is idempotent & F is commutative & F is associative & F is having_a_unity holds
for B1, B2 being Element of Fin X holds F $$ ((B1 \/ B2),f) = F . ((F $$ (B1,f)),(F $$ (B2,f)))

proof end;

theorem :: SETWISEO:34

for X, Y being non empty set
for F being BinOp of Y st F is commutative & F is associative & F is idempotent & F is having_a_unity holds
for f, g being Function of X,Y
for A, B being Element of Fin X st f .: A = g .: B holds
F $$ (A,f) = F $$ (B,g)

proof end;

theorem Th32: :: SETWISEO:35

for A, X, Y being non empty set
for F being BinOp of A st F is idempotent & F is commutative & F is associative & F is having_a_unity holds
for B being Element of Fin X
for f being Function of X,Y
for g being Function of Y,A holds F $$ ((f .: B),g) = F $$ (B,(g * f))

proof end;

theorem :: SETWISEO:36

for X, Y being non empty set
for F being BinOp of Y st F is commutative & F is associative & F is idempotent & F is having_a_unity holds
for Z being non empty set
for G being BinOp of Z st G is commutative & G is associative & G is idempotent & G is having_a_unity holds
for f being Function of X,Y
for g being Function of Y,Z st g . (the_unity_wrt F) = the_unity_wrt G & ( for x, y being Element of Y holds g . (F . (x,y)) = G . ((g . x),(g . y)) ) holds
for B being Element of Fin X holds g . (F $$ (B,f)) = G $$ (B,(g * f))

proof end;

definition

let A be set ;

func FinUnion A -> BinOp of (Fin A) means :Def4: :: SETWISEO:def 4
for x, y being Element of Fin A holds it . (x,y) = x \/ y;

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def4 defines FinUnion SETWISEO:def 4 :

theorem Th34: :: SETWISEO:37

for A being set holds FinUnion A is idempotent

proof end;

theorem Th35: :: SETWISEO:38

for A being set holds FinUnion A is commutative

proof end;

theorem Th36: :: SETWISEO:39

for A being set holds FinUnion A is associative

proof end;

theorem Th37: :: SETWISEO:40

for A being set holds {}. A is_a_unity_wrt FinUnion A

proof end;

theorem Th38: :: SETWISEO:41

for A being set holds FinUnion A is having_a_unity

proof end;

theorem :: SETWISEO:42

for A being set holds the_unity_wrt (FinUnion A) is_a_unity_wrt FinUnion A by Th11, Th38;

theorem Th40: :: SETWISEO:43

for A being set holds the_unity_wrt (FinUnion A) = {}

proof end;

definition

let X be non empty set ;
let A be set ;
let B be Element of Fin X;
let f be Function of X,(Fin A);

func FinUnion (B,f) -> Element of Fin A equals :: SETWISEO:def 5
(FinUnion A) $$ (B,f);

coherence ;

end;

:: deftheorem defines FinUnion SETWISEO:def 5 :

theorem :: SETWISEO:44

for X being non empty set
for A being set
for f being Function of X,(Fin A)
for i being Element of X holds FinUnion ({.i.},f) = f . i

proof end;

theorem :: SETWISEO:45

for X being non empty set
for A being set
for f being Function of X,(Fin A)
for i, j being Element of X holds FinUnion ({.i,j.},f) = (f . i) \/ (f . j)

proof end;

theorem :: SETWISEO:46

for X being non empty set
for A being set
for f being Function of X,(Fin A)
for i, j, k being Element of X holds FinUnion ({.i,j,k.},f) = ((f . i) \/ (f . j)) \/ (f . k)

proof end;

theorem Th44: :: SETWISEO:47

for X being non empty set
for A being set
for f being Function of X,(Fin A) holds FinUnion (({}. X),f) = {}

proof end;

theorem Th45: :: SETWISEO:48

for X being non empty set
for A being set
for f being Function of X,(Fin A)
for i being Element of X
for B being Element of Fin X holds FinUnion ((B \/ {.i.}),f) = (FinUnion (B,f)) \/ (f . i)

proof end;

theorem Th46: :: SETWISEO:49

for X being non empty set
for A being set
for f being Function of X,(Fin A)
for B being Element of Fin X holds FinUnion (B,f) = union (f .: B)

proof end;

theorem :: SETWISEO:50

for X being non empty set
for A being set
for f being Function of X,(Fin A)
for B1, B2 being Element of Fin X holds FinUnion ((B1 \/ B2),f) = (FinUnion (B1,f)) \/ (FinUnion (B2,f))

proof end;

theorem :: SETWISEO:51

for X, Y being non empty set
for A being set
for B being Element of Fin X
for f being Function of X,Y
for g being Function of Y,(Fin A) holds FinUnion ((f .: B),g) = FinUnion (B,(g * f))

proof end;

theorem Th49: :: SETWISEO:52

for A, X being non empty set
for Y being set
for G being BinOp of A st G is commutative & G is associative & G is idempotent holds
for B being Element of Fin X st B <> {} holds
for f being Function of X,(Fin Y)
for g being Function of (Fin Y),A st ( for x, y being Element of Fin Y holds g . (x \/ y) = G . ((g . x),(g . y)) ) holds
g . (FinUnion (B,f)) = G $$ (B,(g * f))

proof end;

theorem Th50: :: SETWISEO:53

for X, Z being non empty set
for Y being set
for G being BinOp of Z st G is commutative & G is associative & G is idempotent & G is having_a_unity holds
for f being Function of X,(Fin Y)
for g being Function of (Fin Y),Z st g . ({}. Y) = the_unity_wrt G & ( for x, y being Element of Fin Y holds g . (x \/ y) = G . ((g . x),(g . y)) ) holds
for B being Element of Fin X holds g . (FinUnion (B,f)) = G $$ (B,(g * f))

proof end;

definition

let A be set ;

func singleton A -> Function of A,(Fin A) means :Def6: :: SETWISEO:def 6
for x being object st x in A holds
it . x = {x};

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def6 defines singleton SETWISEO:def 6 :

theorem Th51: :: SETWISEO:54

for A being non empty set
for f being Function of A,(Fin A) holds
( f = singleton A iff for x being Element of A holds f . x = {x} )

proof end;

theorem Th52: :: SETWISEO:55

for X being non empty set
for x being set
for y being Element of X holds
( x in (singleton X) . y iff x = y )

proof end;

theorem :: SETWISEO:56

for X being non empty set
for x, y, z being Element of X st x in (singleton X) . z & y in (singleton X) . z holds
x = y

proof end;

Lm2: for D being non empty set
for X, P being set
for f being Function of X,D holds f .: P c= D
;

theorem Th54: :: SETWISEO:57

for X being non empty set
for A being set
for f being Function of X,(Fin A)
for B being Element of Fin X
for x being set holds
( x in FinUnion (B,f) iff ex i being Element of X st
( i in B & x in f . i ) )

proof end;

theorem :: SETWISEO:58

for X being non empty set
for B being Element of Fin X holds FinUnion (B,(singleton X)) = B

proof end;

theorem :: SETWISEO:59

for X being non empty set
for Y, Z being set
for f being Function of X,(Fin Y)
for g being Function of (Fin Y),(Fin Z) st g . ({}. Y) = {}. Z & ( for x, y being Element of Fin Y holds g . (x \/ y) = (g . x) \/ (g . y) ) holds
for B being Element of Fin X holds g . (FinUnion (B,f)) = FinUnion (B,(g * f))

proof end;