:: PROB_1 semantic presentation

theorem Th1: :: PROB_1:1
canceled;

theorem Th2: :: PROB_1:2
canceled;

theorem Th3: :: PROB_1:3
for r being real number
for seq being Real_Sequence st ex n being Element of NAT st
for m being Element of NAT st n <= m holds
seq . m = r holds
( seq is convergent & lim seq = r )
proof end;

definition
let X be set ;
let IT be Subset-Family of X;
attr a2 is compl-closed means :Def1: :: PROB_1:def 1
for A being Subset of X st A in IT holds
A ` in IT;
end;

:: deftheorem Def1 defines compl-closed PROB_1:def 1 :
for X being set
for IT being Subset-Family of X holds
( IT is compl-closed iff for A being Subset of X st A in IT holds
A ` in IT );

registration
let X be set ;
cluster non empty cap-closed compl-closed Element of bool (bool a1);
existence
ex b1 being Subset-Family of X st
( not b1 is empty & b1 is compl-closed & b1 is cap-closed )
proof end;
end;

definition
let X be set ;
mode Field_Subset of a1 is non empty cap-closed compl-closed Subset-Family of a1;
end;

theorem Th4: :: PROB_1:4
for X being set
for A, B being Subset of X holds {A,B} is Subset-Family of X
proof end;

theorem Th5: :: PROB_1:5
canceled;

theorem Th6: :: PROB_1:6
for X being set
for F being Field_Subset of X ex A being Subset of X st A in F
proof end;

theorem Th7: :: PROB_1:7
canceled;

theorem Th8: :: PROB_1:8
canceled;

theorem Th9: :: PROB_1:9
for X being set
for F being Field_Subset of X
for A, B being set st A in F & B in F holds
A \/ B in F
proof end;

theorem Th10: :: PROB_1:10
for X being set
for F being Field_Subset of X holds {} in F
proof end;

theorem Th11: :: PROB_1:11
for X being set
for F being Field_Subset of X holds X in F
proof end;

theorem Th12: :: PROB_1:12
for X being set
for F being Field_Subset of X
for A, B being Subset of X st A in F & B in F holds
A \ B in F
proof end;

theorem Th13: :: PROB_1:13
for X being set
for F being Field_Subset of X
for A, B being set st A in F & B in F holds
(A \ B) \/ B in F
proof end;

theorem Th14: :: PROB_1:14
for X being set holds {{} ,X} is Field_Subset of X
proof end;

theorem Th15: :: PROB_1:15
for X being set holds bool X is Field_Subset of X
proof end;

theorem Th16: :: PROB_1:16
for X being set
for F being Field_Subset of X holds
( {{} ,X} c= F & F c= bool X )
proof end;

theorem Th17: :: PROB_1:17
canceled;

theorem Th18: :: PROB_1:18
for X being set holds
( ( for p being set st p in [:NAT ,{X}:] holds
ex x, y being set st [x,y] = p ) & ( for x, y, z being set st [x,y] in [:NAT ,{X}:] & [x,z] in [:NAT ,{X}:] holds
y = z ) )
proof end;

theorem Th19: :: PROB_1:19
for X being set ex f being Function st
( dom f = NAT & ( for n being Element of NAT holds f . n = X ) )
proof end;

definition
let X be set ;
mode SetSequence of a1 is Function of NAT , bool a1;
end;

Lemma58: for X being set
for A1 being SetSequence of X holds
( dom A1 = NAT & ( for n being Element of NAT holds A1 . n in bool X ) )
by FUNCT_2:def 1;

theorem Th20: :: PROB_1:20
canceled;

theorem Th21: :: PROB_1:21
for X being set ex A1 being SetSequence of X st
for n being Element of NAT holds A1 . n = X
proof end;

theorem Th22: :: PROB_1:22
for X being set
for A, B being Subset of X ex A1 being SetSequence of X st
( A1 . 0 = A & ( for n being Element of NAT st n <> 0 holds
A1 . n = B ) )
proof end;

definition
let X be set ;
let A1 be SetSequence of X;
let n be Element of NAT ;
redefine func . as c2 . c3 -> Subset of a1;
coherence
A1 . n is Subset of X
by ;
end;

theorem Th23: :: PROB_1:23
for X being set
for A1 being SetSequence of X holds union (rng A1) is Subset of X
proof end;

definition
let X be set ;
let A1 be SetSequence of X;
redefine func Union as Union c2 -> Subset of a1;
coherence
Union A1 is Subset of X
proof end;
end;

theorem Th24: :: PROB_1:24
canceled;

theorem Th25: :: PROB_1:25
for X, x being set
for A1 being SetSequence of X holds
( x in Union A1 iff ex n being Element of NAT st x in A1 . n )
proof end;

theorem Th26: :: PROB_1:26
for X being set
for A1 being SetSequence of X ex B1 being SetSequence of X st
for n being Element of NAT holds B1 . n = (A1 . n) `
proof end;

definition
let X be set ;
let A1 be SetSequence of X;
canceled;
canceled;
func Complement c2 -> SetSequence of a1 means :Def4: :: PROB_1:def 4
for n being Element of NAT holds it . n = (A1 . n) ` ;
existence
ex b1 being SetSequence of X st
for n being Element of NAT holds b1 . n = (A1 . n) `
by ;
uniqueness
for b1, b2 being SetSequence of X st ( for n being Element of NAT holds b1 . n = (A1 . n) ` ) & ( for n being Element of NAT holds b2 . n = (A1 . n) ` ) holds
b1 = b2
proof end;
involutiveness
for b1, b2 being SetSequence of X st ( for n being Element of NAT holds b1 . n = (b2 . n) ` ) holds
for n being Element of NAT holds b2 . n = (b1 . n) `
proof end;
end;

:: deftheorem Def2 PROB_1:def 2 :
canceled;

:: deftheorem Def3 PROB_1:def 3 :
canceled;

:: deftheorem Def4 defines Complement PROB_1:def 4 :
for X being set
for A1, b3 being SetSequence of X holds
( b3 = Complement A1 iff for n being Element of NAT holds b3 . n = (A1 . n) ` );

definition
let X be set ;
let A1 be SetSequence of X;
func Intersection c2 -> Subset of a1 equals :: PROB_1:def 5
(Union (Complement A1)) ` ;
correctness
coherence
(Union (Complement A1)) ` is Subset of X
;
;
end;

:: deftheorem Def5 defines Intersection PROB_1:def 5 :
for X being set
for A1 being SetSequence of X holds Intersection A1 = (Union (Complement A1)) ` ;

theorem Th27: :: PROB_1:27
canceled;

theorem Th28: :: PROB_1:28
canceled;

theorem Th29: :: PROB_1:29
for X, x being set
for A1 being SetSequence of X holds
( x in Intersection A1 iff for n being Element of NAT holds x in A1 . n )
proof end;

theorem Th30: :: PROB_1:30
for X being set
for A1 being SetSequence of X
for A, B being Subset of X st A1 . 0 = A & ( for n being Element of NAT st n <> 0 holds
A1 . n = B ) holds
Intersection A1 = A /\ B
proof end;

definition
let X be set ;
let A1 be SetSequence of X;
attr a2 is non-increasing means :Def6: :: PROB_1:def 6
for n, m being Element of NAT st n <= m holds
A1 . m c= A1 . n;
attr a2 is non-decreasing means :: PROB_1:def 7
for n, m being Element of NAT st n <= m holds
A1 . n c= A1 . m;
end;

:: deftheorem Def6 defines non-increasing PROB_1:def 6 :
for X being set
for A1 being SetSequence of X holds
( A1 is non-increasing iff for n, m being Element of NAT st n <= m holds
A1 . m c= A1 . n );

:: deftheorem Def7 defines non-decreasing PROB_1:def 7 :
for X being set
for A1 being SetSequence of X holds
( A1 is non-decreasing iff for n, m being Element of NAT st n <= m holds
A1 . n c= A1 . m );

definition
let X be set ;
let F be Subset-Family of X;
attr a2 is sigma-multiplicative means :Def8: :: PROB_1:def 8
for A1 being SetSequence of X st ( for n being Element of NAT holds A1 . n in F ) holds
Intersection A1 in F;
end;

:: deftheorem Def8 defines sigma-multiplicative PROB_1:def 8 :
for X being set
for F being Subset-Family of X holds
( F is sigma-multiplicative iff for A1 being SetSequence of X st ( for n being Element of NAT holds A1 . n in F ) holds
Intersection A1 in F );

registration
let X be set ;
cluster non empty compl-closed sigma-multiplicative Element of bool (bool a1);
existence
ex b1 being Subset-Family of X st
( b1 is compl-closed & b1 is sigma-multiplicative & not b1 is empty )
proof end;
end;

definition
let X be set ;
mode SigmaField of a1 is non empty compl-closed sigma-multiplicative Subset-Family of a1;
end;

theorem Th31: :: PROB_1:31
canceled;

theorem Th32: :: PROB_1:32
for X being set
for S being non empty set holds
( S is SigmaField of X iff ( S c= bool X & ( for A1 being SetSequence of X st ( for n being Element of NAT holds A1 . n in S ) holds
Intersection A1 in S ) & ( for A being Subset of X st A in S holds
A ` in S ) ) ) by , ;

theorem Th33: :: PROB_1:33
canceled;

theorem Th34: :: PROB_1:34
canceled;

theorem Th35: :: PROB_1:35
for Y, X being set st Y is SigmaField of X holds
Y is Field_Subset of X
proof end;

registration
let X be set ;
cluster -> cap-closed Element of bool (bool a1);
coherence
for b1 being SigmaField of X holds
( b1 is cap-closed & b1 is compl-closed )
by ;
end;

theorem Th36: :: PROB_1:36
canceled;

theorem Th37: :: PROB_1:37
canceled;

theorem Th38: :: PROB_1:38
for X being set
for Si being SigmaField of X ex A being Subset of X st A in Si
proof end;

theorem Th39: :: PROB_1:39
canceled;

theorem Th40: :: PROB_1:40
canceled;

theorem Th41: :: PROB_1:41
for X being set
for Si being SigmaField of X
for A, B being Subset of X st A in Si & B in Si holds
A \/ B in Si by ;

theorem Th42: :: PROB_1:42
for X being set
for Si being SigmaField of X holds {} in Si by ;

theorem Th43: :: PROB_1:43
for X being set
for Si being SigmaField of X holds X in Si by ;

theorem Th44: :: PROB_1:44
for X being set
for Si being SigmaField of X
for A, B being Subset of X st A in Si & B in Si holds
A \ B in Si by ;

definition
let X be set ;
let Si be SigmaField of X;
mode SetSequence of c2 -> SetSequence of a1 means :Def9: :: PROB_1:def 9
for n being Element of NAT holds it . n in Si;
existence
ex b1 being SetSequence of X st
for n being Element of NAT holds b1 . n in Si
proof end;
end;

:: deftheorem Def9 defines SetSequence PROB_1:def 9 :
for X being set
for Si being SigmaField of X
for b3 being SetSequence of X holds
( b3 is SetSequence of Si iff for n being Element of NAT holds b3 . n in Si );

theorem Th45: :: PROB_1:45
canceled;

theorem Th46: :: PROB_1:46
for X being set
for Si being SigmaField of X
for ASeq being SetSequence of Si holds Union ASeq in Si
proof end;

definition
let X be set ;
let F be SigmaField of X;
mode Event of c2 -> Subset of a1 means :Def10: :: PROB_1:def 10
it in F;
existence
ex b1 being Subset of X st b1 in F
proof end;
end;

:: deftheorem Def10 defines Event PROB_1:def 10 :
for X being set
for F being SigmaField of X
for b3 being Subset of X holds
( b3 is Event of F iff b3 in F );

theorem Th47: :: PROB_1:47
canceled;

theorem Th48: :: PROB_1:48
for X, x being set
for Si being SigmaField of X st x in Si holds
x is Event of Si by ;

theorem Th49: :: PROB_1:49
for X being set
for Si being SigmaField of X
for A, B being Event of Si holds A /\ B is Event of Si
proof end;

theorem Th50: :: PROB_1:50
for X being set
for Si being SigmaField of X
for A being Event of Si holds A ` is Event of Si
proof end;

theorem Th51: :: PROB_1:51
for X being set
for Si being SigmaField of X
for A, B being Event of Si holds A \/ B is Event of Si
proof end;

theorem Th52: :: PROB_1:52
for X being set
for Si being SigmaField of X holds {} is Event of Si
proof end;

theorem Th53: :: PROB_1:53
for X being set
for Si being SigmaField of X holds X is Event of Si
proof end;

theorem Th54: :: PROB_1:54
for X being set
for Si being SigmaField of X
for A, B being Event of Si holds A \ B is Event of Si
proof end;

registration
let X be set ;
let Si be SigmaField of X;
cluster empty Event of a2;
existence
ex b1 being Event of Si st b1 is empty
proof end;
end;

definition
let X be set ;
let Si be SigmaField of X;
func [#] c2 -> Event of a2 equals :: PROB_1:def 11
X;
coherence
X is Event of Si
by ;
end;

:: deftheorem Def11 defines [#] PROB_1:def 11 :
for X being set
for Si being SigmaField of X holds [#] Si = X;

definition
let X be set ;
let Si be SigmaField of X;
let A be Event of Si, B be Event of Si;
redefine func /\ as c3 /\ c4 -> Event of a2;
coherence
A /\ B is Event of Si
by ;
redefine func \/ as c3 \/ c4 -> Event of a2;
coherence
A \/ B is Event of Si
by ;
redefine func \ as c3 \ c4 -> Event of a2;
coherence
A \ B is Event of Si
by ;
end;

theorem Th55: :: PROB_1:55
canceled;

theorem Th56: :: PROB_1:56
canceled;

theorem Th57: :: PROB_1:57
for Omega being non empty set
for ASeq being SetSequence of Omega
for Sigma being SigmaField of Omega holds
( ASeq is SetSequence of Sigma iff for n being Element of NAT holds ASeq . n is Event of Sigma )
proof end;

theorem Th58: :: PROB_1:58
for Omega being non empty set
for ASeq being SetSequence of Omega
for Sigma being SigmaField of Omega st ASeq is SetSequence of Sigma holds
Union ASeq is Event of Sigma
proof end;

theorem Th59: :: PROB_1:59
for Omega being non empty set
for p being set
for Sigma being SigmaField of Omega ex f being Function st
( dom f = Sigma & ( for D being Subset of Omega st D in Sigma holds
( ( p in D implies f . D = 1 ) & ( not p in D implies f . D = 0 ) ) ) )
proof end;

theorem Th60: :: PROB_1:60
for Omega being non empty set
for p being set
for Sigma being SigmaField of Omega ex P being Function of Sigma, REAL st
for D being Subset of Omega st D in Sigma holds
( ( p in D implies P . D = 1 ) & ( not p in D implies P . D = 0 ) )
proof end;

theorem Th61: :: PROB_1:61
canceled;

theorem Th62: :: PROB_1:62
for Omega being non empty set
for Sigma being SigmaField of Omega
for ASeq being SetSequence of Sigma
for P being Function of Sigma, REAL holds P * ASeq is Real_Sequence
proof end;

definition
let Omega be non empty set ;
let Sigma be SigmaField of Omega;
let ASeq be SetSequence of Sigma;
let P be Function of Sigma, REAL ;
redefine func * as c4 * c3 -> Real_Sequence;
coherence
ASeq * P is Real_Sequence
by Th3;
end;

definition
let Omega be non empty set ;
let Sigma be SigmaField of Omega;
let P be Function of Sigma, REAL ;
let A be Event of Sigma;
redefine func . as c3 . c4 -> Real;
coherence
P . A is Real
proof end;
end;

definition
let Omega be non empty set ;
let Sigma be SigmaField of Omega;
canceled;
mode Probability of c2 -> Function of a2, REAL means :Def13: :: PROB_1:def 13
( ( for A being Event of Sigma holds 0 <= it . A ) & it . Omega = 1 & ( for A, B being Event of Sigma st A misses B holds
it . (A \/ B) = (it . A) + (it . B) ) & ( for ASeq being SetSequence of Sigma st ASeq is non-increasing holds
( it * ASeq is convergent & lim (it * ASeq) = it . (Intersection ASeq) ) ) );
existence
ex b1 being Function of Sigma, REAL st
( ( for A being Event of Sigma holds 0 <= b1 . A ) & b1 . Omega = 1 & ( for A, B being Event of Sigma st A misses B holds
b1 . (A \/ B) = (b1 . A) + (b1 . B) ) & ( for ASeq being SetSequence of Sigma st ASeq is non-increasing holds
( b1 * ASeq is convergent & lim (b1 * ASeq) = b1 . (Intersection ASeq) ) ) )
proof end;
end;

:: deftheorem Def12 PROB_1:def 12 :
canceled;

:: deftheorem Def13 defines Probability PROB_1:def 13 :
for Omega being non empty set
for Sigma being SigmaField of Omega
for b3 being Function of Sigma, REAL holds
( b3 is Probability of Sigma iff ( ( for A being Event of Sigma holds 0 <= b3 . A ) & b3 . Omega = 1 & ( for A, B being Event of Sigma st A misses B holds
b3 . (A \/ B) = (b3 . A) + (b3 . B) ) & ( for ASeq being SetSequence of Sigma st ASeq is non-increasing holds
( b3 * ASeq is convergent & lim (b3 * ASeq) = b3 . (Intersection ASeq) ) ) ) );

theorem Th63: :: PROB_1:63
canceled;

theorem Th64: :: PROB_1:64
for Omega being non empty set
for Sigma being SigmaField of Omega
for P being Probability of Sigma holds P . {} = 0
proof end;

theorem Th65: :: PROB_1:65
canceled;

theorem Th66: :: PROB_1:66
for Omega being non empty set
for Sigma being SigmaField of Omega
for P being Probability of Sigma holds P . ([#] Sigma) = 1 by ;

theorem Th67: :: PROB_1:67
for Omega being non empty set
for Sigma being SigmaField of Omega
for A being Event of Sigma
for P being Probability of Sigma holds (P . (([#] Sigma) \ A)) + (P . A) = 1
proof end;

theorem Th68: :: PROB_1:68
for Omega being non empty set
for Sigma being SigmaField of Omega
for A being Event of Sigma
for P being Probability of Sigma holds P . (([#] Sigma) \ A) = 1 - (P . A)
proof end;

theorem Th69: :: PROB_1:69
for Omega being non empty set
for Sigma being SigmaField of Omega
for A, B being Event of Sigma
for P being Probability of Sigma st A c= B holds
P . (B \ A) = (P . B) - (P . A)
proof end;

theorem Th70: :: PROB_1:70
for Omega being non empty set
for Sigma being SigmaField of Omega
for A, B being Event of Sigma
for P being Probability of Sigma st A c= B holds
P . A <= P . B
proof end;

theorem Th71: :: PROB_1:71
for Omega being non empty set
for Sigma being SigmaField of Omega
for A being Event of Sigma
for P being Probability of Sigma holds P . A <= 1
proof end;

theorem Th72: :: PROB_1:72
for Omega being non empty set
for Sigma being SigmaField of Omega
for A, B being Event of Sigma
for P being Probability of Sigma holds P . (A \/ B) = (P . A) + (P . (B \ A))
proof end;

theorem Th73: :: PROB_1:73
for Omega being non empty set
for Sigma being SigmaField of Omega
for A, B being Event of Sigma
for P being Probability of Sigma holds P . (A \/ B) = (P . A) + (P . (B \ (A /\ B)))
proof end;

theorem Th74: :: PROB_1:74
for Omega being non empty set
for Sigma being SigmaField of Omega
for A, B being Event of Sigma
for P being Probability of Sigma holds P . (A \/ B) = ((P . A) + (P . B)) - (P . (A /\ B))
proof end;

theorem Th75: :: PROB_1:75
for Omega being non empty set
for Sigma being SigmaField of Omega
for A, B being Event of Sigma
for P being Probability of Sigma holds P . (A \/ B) <= (P . A) + (P . B)
proof end;

theorem Th76: :: PROB_1:76
for Omega being non empty set holds bool Omega is SigmaField of Omega
proof end;

Lemma120: for Omega being non empty set
for X being Subset-Family of Omega ex Y being SigmaField of Omega st
( X c= Y & ( for Z being set st X c= Z & Z is SigmaField of Omega holds
Y c= Z ) )
proof end;

definition
let Omega be non empty set ;
let X be Subset-Family of Omega;
func sigma c2 -> SigmaField of a1 means :: PROB_1:def 14
( X c= it & ( for Z being set st X c= Z & Z is SigmaField of Omega holds
it c= Z ) );
existence
ex b1 being SigmaField of Omega st
( X c= b1 & ( for Z being set st X c= Z & Z is SigmaField of Omega holds
b1 c= Z ) )
by ;
uniqueness
for b1, b2 being SigmaField of Omega st X c= b1 & ( for Z being set st X c= Z & Z is SigmaField of Omega holds
b1 c= Z ) & X c= b2 & ( for Z being set st X c= Z & Z is SigmaField of Omega holds
b2 c= Z ) holds
b1 = b2
proof end;
end;

:: deftheorem Def14 defines sigma PROB_1:def 14 :
for Omega being non empty set
for X being Subset-Family of Omega
for b3 being SigmaField of Omega holds
( b3 = sigma X iff ( X c= b3 & ( for Z being set st X c= Z & Z is SigmaField of Omega holds
b3 c= Z ) ) );

definition
let r be real number ;
func halfline c1 -> Subset of REAL equals :: PROB_1:def 15
{ r1 where r1 is Element of REAL : r1 < r } ;
coherence
{ r1 where r1 is Element of REAL : r1 < r } is Subset of REAL
proof end;
end;

:: deftheorem Def15 defines halfline PROB_1:def 15 :
for r being real number holds halfline r = { r1 where r1 is Element of REAL : r1 < r } ;

definition
func Family_of_halflines -> Subset-Family of REAL equals :: PROB_1:def 16
{ D where D is Subset of REAL : ex r being real number st D = halfline r } ;
coherence
{ D where D is Subset of REAL : ex r being real number st D = halfline r } is Subset-Family of REAL
proof end;
end;

:: deftheorem Def16 defines Family_of_halflines PROB_1:def 16 :
Family_of_halflines = { D where D is Subset of REAL : ex r being real number st D = halfline r } ;

definition
func Borel_Sets -> SigmaField of REAL equals :: PROB_1:def 17
sigma Family_of_halflines ;
correctness
coherence
sigma Family_of_halflines is SigmaField of REAL
;
;
end;

:: deftheorem Def17 defines Borel_Sets PROB_1:def 17 :
Borel_Sets = sigma Family_of_halflines ;