:: Properties of the Intervals of Real Numbers
:: by J\'ozef Bia{\l}as
::
:: Received January 12, 1993
:: Copyright (c) 1993-2011 Association of Mizar Users

begin

theorem :: MEASURE5:1

canceled;

theorem :: MEASURE5:2

canceled;

theorem :: MEASURE5:3

canceled;

theorem :: MEASURE5:4

canceled;

theorem :: MEASURE5:5

canceled;

theorem :: MEASURE5:6

canceled;

theorem :: MEASURE5:7

canceled;

theorem :: MEASURE5:8

canceled;

scheme :: MEASURE5:sch 1
RSetEq{ P₁[ set ] } :

for X1, X2 being Subset of REAL st ( for x being R_eal holds
( x in X1 iff P₁[x] ) ) & ( for x being R_eal holds
( x in X2 iff P₁[x] ) ) holds
X1 = X2

proof end;

definition

canceled;
let a, b be R_eal;
:: original: ].
redefine func ].a,b.[ -> Subset of REAL means :: MEASURE5:def 2
for x being R_eal holds
( x in it iff ( a < x & x < b ) );
coherence
].a,b.[ is Subset of REAL

proof end;

compatibility
for b₁ being Subset of REAL holds
( b₁ = ].a,b.[ iff for x being R_eal holds
( x in b₁ iff ( a < x & x < b ) ) )

proof end;

end;

:: deftheorem MEASURE5:def 1 :
canceled;

:: deftheorem defines ]. MEASURE5:def 2 :
for a, b being R_eal
for b₃ being Subset of REAL holds
( b₃ = ].a,b.[ iff for x being R_eal holds
( x in b₃ iff ( a < x & x < b ) ) );

definition

canceled;
canceled;
let IT be Subset of REAL;
attr IT is open_interval means :Def5: :: MEASURE5:def 5
ex a, b being R_eal st IT = ].a,b.[;
attr IT is closed_interval means :Def6: :: MEASURE5:def 6
ex a, b being real number st IT = [.a,b.];

end;

:: deftheorem MEASURE5:def 3 :
canceled;

:: deftheorem MEASURE5:def 4 :
canceled;

:: deftheorem Def5 defines open_interval MEASURE5:def 5 :
for IT being Subset of REAL holds
( IT is open_interval iff ex a, b being R_eal st IT = ].a,b.[ );

:: deftheorem Def6 defines closed_interval MEASURE5:def 6 :
for IT being Subset of REAL holds
( IT is closed_interval iff ex a, b being real number st IT = [.a,b.] );

registration

cluster non empty complex-membered ext-real-membered real-membered open_interval Element of K18(REAL);
existence
ex b₁ being Subset of REAL st
( not b₁ is empty & b₁ is open_interval )

proof end;

cluster non empty complex-membered ext-real-membered real-membered closed_interval Element of K18(REAL);
existence
ex b₁ being Subset of REAL st
( not b₁ is empty & b₁ is closed_interval )

proof end;

end;

definition

let IT be Subset of REAL;
attr IT is right_open_interval means :Def7: :: MEASURE5:def 7
ex a being real number ex b being R_eal st IT = [.a,b.[;

end;

:: deftheorem Def7 defines right_open_interval MEASURE5:def 7 :
for IT being Subset of REAL holds
( IT is right_open_interval iff ex a being real number ex b being R_eal st IT = [.a,b.[ );

notation

let IT be Subset of REAL;
synonym left_closed_interval IT for right_open_interval ;

end;

definition

let IT be Subset of REAL;
attr IT is left_open_interval means :Def8: :: MEASURE5:def 8
ex a being R_eal ex b being real number st IT = ].a,b.];

end;

:: deftheorem Def8 defines left_open_interval MEASURE5:def 8 :
for IT being Subset of REAL holds
( IT is left_open_interval iff ex a being R_eal ex b being real number st IT = ].a,b.] );

notation

let IT be Subset of REAL;
synonym right_closed_interval IT for left_open_interval ;

end;

registration

cluster non empty complex-membered ext-real-membered real-membered right_open_interval Element of K18(REAL);
existence
ex b₁ being Subset of REAL st
( not b₁ is empty & b₁ is right_open_interval )

proof end;

cluster non empty complex-membered ext-real-membered real-membered left_open_interval Element of K18(REAL);
existence
ex b₁ being Subset of REAL st
( not b₁ is empty & b₁ is left_open_interval )

proof end;

end;

definition

mode Interval is interval Subset of REAL;

end;

registration

cluster open_interval -> interval Element of K18(REAL);
coherence
for b₁ being Subset of REAL st b₁ is open_interval holds
b₁ is interval

proof end;

cluster closed_interval -> interval Element of K18(REAL);
coherence
for b₁ being Subset of REAL st b₁ is closed_interval holds
b₁ is interval

proof end;

cluster right_open_interval -> interval Element of K18(REAL);
coherence
for b₁ being Subset of REAL st b₁ is right_open_interval holds
b₁ is interval

proof end;

cluster left_open_interval -> interval Element of K18(REAL);
coherence
for b₁ being Subset of REAL st b₁ is left_open_interval holds
b₁ is interval

proof end;

end;

theorem :: MEASURE5:9

for I being interval Subset of REAL holds
( I is open_interval or I is closed_interval or I is right_open_interval or I is left_open_interval )

proof end;

theorem :: MEASURE5:10

canceled;

theorem :: MEASURE5:11

canceled;

theorem :: MEASURE5:12

canceled;

theorem :: MEASURE5:13

canceled;

theorem :: MEASURE5:14

canceled;

theorem :: MEASURE5:15

canceled;

theorem :: MEASURE5:16

canceled;

theorem Th17: :: MEASURE5:17

for a, b being R_eal st a < b holds
ex x being R_eal st
( a < x & x < b & x in REAL )

proof end;

theorem :: MEASURE5:18

for a, b, c being R_eal st a < b & a < c holds
ex x being R_eal st
( a < x & x < b & x < c & x in REAL )

proof end;

theorem :: MEASURE5:19

for a, b, c being R_eal st a < c & b < c holds
ex x being R_eal st
( a < x & b < x & x < c & x in REAL )

proof end;

definition

canceled;
let A be ext-real-membered set ;
func diameter A -> R_eal equals :Def10: :: MEASURE5:def 10
(sup A) - (inf A) if A <> {}
otherwise 0. ;
coherence
( ( A <> {} implies (sup A) - (inf A) is R_eal ) & ( not A <> {} implies 0. is R_eal ) ) ;
consistency
for b₁ being R_eal holds verum ;

end;

:: deftheorem MEASURE5:def 9 :
canceled;

:: deftheorem Def10 defines diameter MEASURE5:def 10 :
for A being ext-real-membered set holds
( ( A <> {} implies diameter A = (sup A) - (inf A) ) & ( not A <> {} implies diameter A = 0. ) );

theorem :: MEASURE5:20

canceled;

theorem :: MEASURE5:21

canceled;

theorem :: MEASURE5:22

canceled;

theorem :: MEASURE5:23

canceled;

theorem :: MEASURE5:24

canceled;

theorem :: MEASURE5:25

canceled;

theorem :: MEASURE5:26

canceled;

theorem :: MEASURE5:27

canceled;

theorem :: MEASURE5:28

canceled;

theorem :: MEASURE5:29

canceled;

theorem :: MEASURE5:30

canceled;

theorem :: MEASURE5:31

canceled;

theorem :: MEASURE5:32

canceled;

theorem :: MEASURE5:33

canceled;

theorem :: MEASURE5:34

canceled;

theorem :: MEASURE5:35

canceled;

theorem :: MEASURE5:36

canceled;

theorem :: MEASURE5:37

canceled;

theorem :: MEASURE5:38

canceled;

theorem :: MEASURE5:39

canceled;

theorem :: MEASURE5:40

canceled;

theorem :: MEASURE5:41

canceled;

theorem :: MEASURE5:42

canceled;

theorem :: MEASURE5:43

canceled;

theorem :: MEASURE5:44

canceled;

theorem :: MEASURE5:45

canceled;

theorem :: MEASURE5:46

canceled;

theorem :: MEASURE5:47

canceled;

theorem :: MEASURE5:48

canceled;

theorem :: MEASURE5:49

canceled;

theorem :: MEASURE5:50

canceled;

theorem :: MEASURE5:51

canceled;

theorem :: MEASURE5:52

canceled;

theorem :: MEASURE5:53

canceled;

theorem :: MEASURE5:54

for a, b being R_eal holds
( ( a < b implies diameter ].a,b.[ = b - a ) & ( b <= a implies diameter ].a,b.[ = 0. ) )

proof end;

theorem :: MEASURE5:55

for a, b being R_eal holds
( ( a <= b implies diameter [.a,b.] = b - a ) & ( b < a implies diameter [.a,b.] = 0. ) )

proof end;

theorem :: MEASURE5:56

for a, b being R_eal holds
( ( a < b implies diameter [.a,b.[ = b - a ) & ( b <= a implies diameter [.a,b.[ = 0. ) )

proof end;

theorem :: MEASURE5:57

for a, b being R_eal holds
( ( a < b implies diameter ].a,b.] = b - a ) & ( b <= a implies diameter ].a,b.] = 0. ) )

proof end;

theorem :: MEASURE5:58

canceled;

theorem :: MEASURE5:59

for A being Interval
for a, b being R_eal st a = -infty & b = +infty & ( A = ].a,b.[ or A = [.a,b.] or A = [.a,b.[ or A = ].a,b.] ) holds
diameter A = +infty

proof end;

registration

cluster empty -> open_interval Element of K18(REAL);
coherence
for b₁ being Subset of REAL st b₁ is empty holds
b₁ is open_interval

proof end;

end;

theorem :: MEASURE5:60

diameter {} = 0. by Def10;

Lm1: for A being Interval holds diameter A >= 0

proof end;

Lm2: for A, B being Interval st A c= B holds
diameter A <= diameter B

proof end;

theorem :: MEASURE5:61

for a, b being R_eal
for A, B being Interval st A c= B & B = [.a,b.] & b <= a holds
( diameter A = 0. & diameter B = 0. )

proof end;

theorem :: MEASURE5:62

for A, B being Interval st A c= B holds
diameter A <= diameter B by Lm2;

theorem :: MEASURE5:63

for A being Interval holds 0. <= diameter A by Lm1;