:: YELLOW15 semantic presentation

scheme :: YELLOW15:sch 1
SeqLambda1C{ F₁() -> Nat, F₂() -> non empty set , P₁[ set ], F₃( set ) -> set , F₄( set ) -> set } :

ex p being FinSequence of F₂() st
( len p = F₁() & ( for i being Nat st i in Seg F₁() holds
( ( P₁[i] implies p . i = F₃(i) ) & ( P₁[i] implies p . i = F₄(i) ) ) ) )

provided

A1: for i being Nat st i in Seg F₁() holds
( ( P₁[i] implies F₃(i) in F₂() ) & ( P₁[i] implies F₄(i) in F₂() ) )

proof end;

definition

let X be set ;
let p be FinSequence of bool X;
:: original: rng
redefine func rng p -> Subset-Family of X;
coherence by FINSEQ_1:def 4;

end;

registration

cluster BOOLEAN -> finite ;
coherence by MARGREL1:def 12;

end;

theorem :: YELLOW15:1

canceled;

theorem Th2: :: YELLOW15:2

for i being Nat
for D being finite set holds i -tuples_on D is finite

proof end;

theorem Th3: :: YELLOW15:3

for T being finite set
for S being Subset-Family of T holds S is finite

proof end;

registration

let T be finite set ;
cluster -> finite Element of bool (bool T);
coherence by Th3;

end;

registration

let T be finite 1-sorted ;
cluster -> finite Element of bool (bool the carrier of T);
coherence

proof end;

end;

theorem Th4: :: YELLOW15:4

for X being non trivial set
for x being Element of X ex y being set st
( y in X & x <> y )

proof end;

definition

let X be set ;
let p be FinSequence of bool X;
let q be FinSequence of BOOLEAN ;
func MergeSequence p,q -> FinSequence of bool X means :Def1: :: YELLOW15:def 1
( len it = len p & ( for i being Nat st i in dom p holds
it . i = IFEQ (q . i),TRUE ,(p . i),(X \ (p . i)) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines MergeSequence YELLOW15:def 1 :

theorem Th5: :: YELLOW15:5

for X being set
for p being FinSequence of bool X
for q being FinSequence of BOOLEAN holds dom (MergeSequence p,q) = dom p

proof end;

theorem Th6: :: YELLOW15:6

for X being set
for p being FinSequence of bool X
for q being FinSequence of BOOLEAN
for i being Nat st q . i = TRUE holds
(MergeSequence p,q) . i = p . i

proof end;

theorem Th7: :: YELLOW15:7

for X being set
for p being FinSequence of bool X
for q being FinSequence of BOOLEAN
for i being Nat st i in dom p & q . i = FALSE holds
(MergeSequence p,q) . i = X \ (p . i)

proof end;

theorem :: YELLOW15:8

for X being set
for q being FinSequence of BOOLEAN holds len (MergeSequence (<*> (bool X)),q) = 0

proof end;

theorem Th9: :: YELLOW15:9

for X being set
for q being FinSequence of BOOLEAN holds MergeSequence (<*> (bool X)),q = <*> (bool X)

proof end;

theorem :: YELLOW15:10

for X being set
for x being Subset of X
for q being FinSequence of BOOLEAN holds len (MergeSequence <*x*>,q) = 1

proof end;

theorem :: YELLOW15:11

for X being set
for x being Subset of X
for q being FinSequence of BOOLEAN holds
( ( q . 1 = TRUE implies (MergeSequence <*x*>,q) . 1 = x ) & ( q . 1 = FALSE implies (MergeSequence <*x*>,q) . 1 = X \ x ) )

proof end;

theorem :: YELLOW15:12

for X being set
for x, y being Subset of X
for q being FinSequence of BOOLEAN holds len (MergeSequence <*x,y*>,q) = 2

proof end;

theorem :: YELLOW15:13

for X being set
for x, y being Subset of X
for q being FinSequence of BOOLEAN holds
( ( q . 1 = TRUE implies (MergeSequence <*x,y*>,q) . 1 = x ) & ( q . 1 = FALSE implies (MergeSequence <*x,y*>,q) . 1 = X \ x ) & ( q . 2 = TRUE implies (MergeSequence <*x,y*>,q) . 2 = y ) & ( q . 2 = FALSE implies (MergeSequence <*x,y*>,q) . 2 = X \ y ) )

proof end;

theorem :: YELLOW15:14

for X being set
for x, y, z being Subset of X
for q being FinSequence of BOOLEAN holds len (MergeSequence <*x,y,z*>,q) = 3

proof end;

theorem :: YELLOW15:15

for X being set
for x, y, z being Subset of X
for q being FinSequence of BOOLEAN holds
( ( q . 1 = TRUE implies (MergeSequence <*x,y,z*>,q) . 1 = x ) & ( q . 1 = FALSE implies (MergeSequence <*x,y,z*>,q) . 1 = X \ x ) & ( q . 2 = TRUE implies (MergeSequence <*x,y,z*>,q) . 2 = y ) & ( q . 2 = FALSE implies (MergeSequence <*x,y,z*>,q) . 2 = X \ y ) & ( q . 3 = TRUE implies (MergeSequence <*x,y,z*>,q) . 3 = z ) & ( q . 3 = FALSE implies (MergeSequence <*x,y,z*>,q) . 3 = X \ z ) )

proof end;

theorem Th16: :: YELLOW15:16

for X being set
for p being FinSequence of bool X holds { (Intersect (rng (MergeSequence p,q))) where q is FinSequence of BOOLEAN : len q = len p } is Subset-Family of X

proof end;

registration

cluster -> boolean-valued FinSequence of BOOLEAN ;
coherence

proof end;

end;

definition

let X be set ;
let Y be finite Subset-Family of X;
func Components Y -> Subset-Family of X means :Def2: :: YELLOW15:def 2
ex p being FinSequence of bool X st
( len p = card Y & rng p = Y & it = { (Intersect (rng (MergeSequence p,q))) where q is FinSequence of BOOLEAN : len q = len p } );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines Components YELLOW15:def 2 :

registration

let X be set ;
let Y be finite Subset-Family of X;
cluster Components Y -> finite ;
coherence

proof end;

end;

theorem Th17: :: YELLOW15:17

for X being set
for Y being empty Subset-Family of X holds Components Y = {X}

proof end;

theorem :: YELLOW15:18

for X being set
for Y, Z being finite Subset-Family of X st Z c= Y holds
Components Y is_finer_than Components Z

proof end;

theorem Th19: :: YELLOW15:19

for X being set
for Y being finite Subset-Family of X holds union (Components Y) = X

proof end;

theorem Th20: :: YELLOW15:20

for X being set
for Y being finite Subset-Family of X
for A, B being set st A in Components Y & B in Components Y & A <> B holds
A misses B

proof end;

definition

let X be set ;
let Y be finite Subset-Family of X;
attr Y is in_general_position means :Def3: :: YELLOW15:def 3
not {} in Components Y;

end;

:: deftheorem Def3 defines in_general_position YELLOW15:def 3 :

theorem :: YELLOW15:21

for X being set
for Y, Z being finite Subset-Family of X st Z is in_general_position & Y c= Z holds
Y is in_general_position

proof end;

theorem :: YELLOW15:22

for X being non empty set
for Y being empty Subset-Family of X holds Y is in_general_position

proof end;

theorem :: YELLOW15:23

for X being non empty set
for Y being finite Subset-Family of X st Y is in_general_position holds
Components Y is a_partition of X

proof end;

theorem Th24: :: YELLOW15:24

for L being non empty RelStr holds
( [#] L is infs-closed & [#] L is sups-closed )

proof end;

theorem Th25: :: YELLOW15:25

for L being non empty RelStr holds
( [#] L is with_bottom & [#] L is with_top )

proof end;

registration

let L be non empty RelStr ;
cluster [#] L -> infs-closed sups-closed with_bottom with_top ;
coherence by Th24, Th25;

end;

theorem :: YELLOW15:26

for L being continuous sup-Semilattice holds [#] L is CLbasis of L

proof end;

theorem :: YELLOW15:27

for L being non empty up-complete Poset st L is finite holds
the carrier of L = the carrier of (CompactSublatt L)

proof end;

theorem :: YELLOW15:28

for L being lower-bounded sup-Semilattice
for B being Subset of L st not B is finite holds
Card B = Card (finsups B)

proof end;

theorem :: YELLOW15:29

for T being non empty T_0 TopSpace holds Card the carrier of T c= Card the topology of T

proof end;

theorem Th30: :: YELLOW15:30

for T being TopStruct
for X being Subset of T st X is open holds
for B being finite Subset-Family of T st B is Basis of T holds
for Y being set holds
( not Y in Components B or X misses Y or Y c= X )

proof end;

theorem :: YELLOW15:31

for T being T_0 TopSpace st not T is finite holds
for B being Basis of T holds not B is finite

proof end;

theorem :: YELLOW15:32

for T being non empty TopSpace st T is finite holds
for B being Basis of T
for x being Element of T holds meet { A where A is Element of the topology of T : x in A } in B

proof end;