Article PROB_2, MML version 4.99.1005
:: PROB_2:th 4
theorem
for b1, b2, b3, b4 being Element of REAL
st b1 <> 0 & b2 <> 0
holds b4 / b2 = b3 / b1
iff
b4 * b1 = b3 * b2;
:: PROB_2:th 5
theorem
for b1 being Element of REAL
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
st b2 is convergent &
(for b4 being Element of NAT holds
b3 . b4 = b1 - (b2 . b4))
holds b3 is convergent & lim b3 = b1 - lim b2;
:: PROB_2:funcnot 1 => PROB_2:func 1
definition
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
let a4 be Element of NAT;
redefine func a3 . a4 -> Event of a2;
end;
:: PROB_2:funcnot 2 => PROB_2:func 2
definition
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
func @Intersection A3 -> Event of a2 equals
Intersection a3;
end;
:: PROB_2:def 1
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
@Intersection b3 = Intersection b3;
:: PROB_2:th 9
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2
for b4 being Event of b2 holds
ex b5 being SetSequence of b2 st
for b6 being Element of NAT holds
b5 . b6 = (b3 . b6) /\ b4;
:: PROB_2:th 10
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being SetSequence of b2
for b5 being Event of b2
st b3 is non-increasing(b1) &
(for b6 being Element of NAT holds
b4 . b6 = (b3 . b6) /\ b5)
holds b4 is non-increasing(b1);
:: PROB_2:th 11
theorem
for b1 being non empty set
for b2 being Element of NAT
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being SetSequence of b3
for b5 being Function-like quasi_total Relation of b3,REAL holds
(b5 * b4) . b2 = b5 . (b4 . b2);
:: PROB_2:th 12
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being SetSequence of b2
for b5 being Event of b2
st for b6 being Element of NAT holds
b3 . b6 = (b4 . b6) /\ b5
holds (Intersection b4) /\ b5 = Intersection b3;
:: PROB_2:th 13
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Probability of b2
st for b5 being Event of b2 holds
b3 . b5 = b4 . b5
holds b3 = b4;
:: PROB_2:th 14
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 is non-increasing(b1)
iff
for b3 being Element of NAT holds
b2 . (b3 + 1) c= b2 . b3;
:: PROB_2:th 15
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 is non-decreasing(b1)
iff
for b3 being Element of NAT holds
b2 . b3 c= b2 . (b3 + 1);
:: PROB_2:th 16
theorem
for b1 being non empty set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
st for b4 being Element of NAT holds
b2 . b4 = b3 . b4
holds b2 = b3;
:: PROB_2:th 17
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 is non-increasing(b1)
iff
Complement b2 is non-decreasing(b1);
:: PROB_2:funcnot 3 => PROB_2:func 3
definition
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
func @Complement A3 -> SetSequence of a2 equals
Complement a3;
end;
:: PROB_2:def 2
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
@Complement b3 = Complement b3;
:: PROB_2:attrnot 1 => PROB_2:attr 1
definition
let a1 be Relation-like Function-like set;
attr a1 is disjoint_valued means
for b1, b2 being set
st b1 <> b2
holds a1 . b1 misses a1 . b2;
end;
:: PROB_2:dfs 3
definiens
let a1 be Relation-like Function-like set;
To prove
a1 is disjoint_valued
it is sufficient to prove
thus for b1, b2 being set
st b1 <> b2
holds a1 . b1 misses a1 . b2;
:: PROB_2:def 3
theorem
for b1 being Relation-like Function-like set holds
b1 is disjoint_valued
iff
for b2, b3 being set
st b2 <> b3
holds b1 . b2 misses b1 . b3;
:: PROB_2:attrnot 2 => PROB_2:attr 2
definition
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
redefine attr a3 is disjoint_valued means
for b1, b2 being Element of NAT
st b1 <> b2
holds a3 . b1 misses a3 . b2;
end;
:: PROB_2:dfs 4
definiens
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
To prove
a1 is disjoint_valued
it is sufficient to prove
thus for b1, b2 being Element of NAT
st b1 <> b2
holds a3 . b1 misses a3 . b2;
:: PROB_2:def 4
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
b3 is disjoint_valued
iff
for b4, b5 being Element of NAT
st b4 <> b5
holds b3 . b4 misses b3 . b5;
:: PROB_2:th 20
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of b2,REAL holds
b3 is Probability of b2
iff
(for b4 being Event of b2 holds
0 <= b3 . b4) &
b3 . b1 = 1 &
(for b4, b5 being Event of b2
st b4 misses b5
holds b3 . (b4 \/ b5) = (b3 . b4) + (b3 . b5)) &
(for b4 being SetSequence of b2
st b4 is non-decreasing(b1)
holds b3 * b4 is convergent & lim (b3 * b4) = b3 . Union b4);
:: PROB_2:th 21
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5, b6 being Event of b2 holds
b3 . ((b4 \/ b5) \/ b6) = ((((b3 . b4) + (b3 . b5)) + (b3 . b6)) - (((b3 . (b4 /\ b5)) + (b3 . (b5 /\ b6))) + (b3 . (b4 /\ b6)))) + (b3 . ((b4 /\ b5) /\ b6));
:: PROB_2:th 22
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5 being Event of b2 holds
b3 . (b4 \ (b4 /\ b5)) = (b3 . b4) - (b3 . (b4 /\ b5));
:: PROB_2:th 23
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5 being Event of b2 holds
b3 . (b4 /\ b5) <= b3 . b5 & b3 . (b4 /\ b5) <= b3 . b4;
:: PROB_2:th 24
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5, b6 being Event of b2
st b4 = b5 `
holds b3 . b6 = (b3 . (b6 /\ b5)) + (b3 . (b6 /\ b4));
:: PROB_2:th 25
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5 being Event of b2 holds
((b3 . b4) + (b3 . b5)) - 1 <= b3 . (b4 /\ b5);
:: PROB_2:th 26
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4 being Event of b2 holds
b3 . b4 = 1 - (b3 . (([#] b2) \ b4));
:: PROB_2:th 27
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4 being Event of b2 holds
b3 . b4 < 1
iff
0 < b3 . (([#] b2) \ b4);
:: PROB_2:th 28
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4 being Event of b2 holds
b3 . (([#] b2) \ b4) < 1
iff
0 < b3 . b4;
:: PROB_2:prednot 1 => PROB_2:pred 1
definition
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be Probability of a2;
let a4, a5 be Event of a2;
pred A4,A5 are_independent_respect_to A3 means
a3 . (a4 /\ a5) = (a3 . a4) * (a3 . a5);
end;
:: PROB_2:dfs 5
definiens
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be Probability of a2;
let a4, a5 be Event of a2;
To prove
a4,a5 are_independent_respect_to a3
it is sufficient to prove
thus a3 . (a4 /\ a5) = (a3 . a4) * (a3 . a5);
:: PROB_2:def 5
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5 being Event of b2 holds
b4,b5 are_independent_respect_to b3
iff
b3 . (b4 /\ b5) = (b3 . b4) * (b3 . b5);
:: PROB_2:prednot 2 => PROB_2:pred 2
definition
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be Probability of a2;
let a4, a5, a6 be Event of a2;
pred A4,A5,A6 are_independent_respect_to A3 means
a3 . ((a4 /\ a5) /\ a6) = ((a3 . a4) * (a3 . a5)) * (a3 . a6) &
a3 . (a4 /\ a5) = (a3 . a4) * (a3 . a5) &
a3 . (a4 /\ a6) = (a3 . a4) * (a3 . a6) &
a3 . (a5 /\ a6) = (a3 . a5) * (a3 . a6);
end;
:: PROB_2:dfs 6
definiens
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be Probability of a2;
let a4, a5, a6 be Event of a2;
To prove
a4,a5,a6 are_independent_respect_to a3
it is sufficient to prove
thus a3 . ((a4 /\ a5) /\ a6) = ((a3 . a4) * (a3 . a5)) * (a3 . a6) &
a3 . (a4 /\ a5) = (a3 . a4) * (a3 . a5) &
a3 . (a4 /\ a6) = (a3 . a4) * (a3 . a6) &
a3 . (a5 /\ a6) = (a3 . a5) * (a3 . a6);
:: PROB_2:def 6
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5, b6 being Event of b2 holds
b4,b5,b6 are_independent_respect_to b3
iff
b3 . ((b4 /\ b5) /\ b6) = ((b3 . b4) * (b3 . b5)) * (b3 . b6) &
b3 . (b4 /\ b5) = (b3 . b4) * (b3 . b5) &
b3 . (b4 /\ b6) = (b3 . b4) * (b3 . b6) &
b3 . (b5 /\ b6) = (b3 . b5) * (b3 . b6);
:: PROB_2:th 31
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5 being Event of b2
st b4,b5 are_independent_respect_to b3
holds b5,b4 are_independent_respect_to b3;
:: PROB_2:th 32
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5, b6 being Event of b2 holds
b4,b5,b6 are_independent_respect_to b3
iff
b3 . ((b4 /\ b5) /\ b6) = ((b3 . b4) * (b3 . b5)) * (b3 . b6) &
b4,b5 are_independent_respect_to b3 &
b5,b6 are_independent_respect_to b3 &
b4,b6 are_independent_respect_to b3;
:: PROB_2:th 33
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5, b6 being Event of b2
st b4,b5,b6 are_independent_respect_to b3
holds b5,b4,b6 are_independent_respect_to b3;
:: PROB_2:th 34
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5, b6 being Event of b2
st b4,b5,b6 are_independent_respect_to b3
holds b4,b6,b5 are_independent_respect_to b3;
:: PROB_2:th 35
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5 being Event of b2
st b5 = {}
holds b4,b5 are_independent_respect_to b3;
:: PROB_2:th 36
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4 being Event of b2 holds
b4,[#] b2 are_independent_respect_to b3;
:: PROB_2:th 37
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Event of b2
for b5 being Probability of b2
st b3,b4 are_independent_respect_to b5
holds b3,([#] b2) \ b4 are_independent_respect_to b5;
:: PROB_2:th 38
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5 being Event of b2
st b4,b5 are_independent_respect_to b3
holds ([#] b2) \ b4,([#] b2) \ b5 are_independent_respect_to b3;
:: PROB_2:th 39
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4, b5 being Event of b2
for b6 being Probability of b2
st b3,b4 are_independent_respect_to b6 & b3,b5 are_independent_respect_to b6 & b4 misses b5
holds b3,b4 \/ b5 are_independent_respect_to b6;
:: PROB_2:th 40
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5 being Event of b2
st b4,b5 are_independent_respect_to b3 & b3 . b4 < 1 & b3 . b5 < 1
holds b3 . (b4 \/ b5) < 1;
:: PROB_2:funcnot 4 => PROB_2:func 4
definition
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be Probability of a2;
let a4 be Event of a2;
assume 0 < a3 . a4;
func A3 .|. A4 -> Probability of a2 means
for b1 being Event of a2 holds
it . b1 = (a3 . (b1 /\ a4)) / (a3 . a4);
end;
:: PROB_2:def 7
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4 being Event of b2
st 0 < b3 . b4
for b5 being Probability of b2 holds
b5 = b3 .|. b4
iff
for b6 being Event of b2 holds
b5 . b6 = (b3 . (b6 /\ b4)) / (b3 . b4);
:: PROB_2:th 42
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5 being Event of b2
st 0 < b3 . b4
holds b3 . (b5 /\ b4) = ((b3 .|. b4) . b5) * (b3 . b4);
:: PROB_2:th 43
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5, b6 being Event of b2
st 0 < b3 . (b4 /\ b5)
holds b3 . ((b4 /\ b5) /\ b6) = ((b3 . b4) * ((b3 .|. b4) . b5)) * ((b3 .|. (b4 /\ b5)) . b6);
:: PROB_2:th 44
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5, b6 being Event of b2
st b6 = b5 ` & 0 < b3 . b5 & 0 < b3 . b6
holds b3 . b4 = (((b3 .|. b5) . b4) * (b3 . b5)) + (((b3 .|. b6) . b4) * (b3 . b6));
:: PROB_2:th 45
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5, b6, b7 being Event of b2
st b5 misses b6 & b7 = (b5 \/ b6) ` & 0 < b3 . b5 & 0 < b3 . b6 & 0 < b3 . b7
holds b3 . b4 = ((((b3 .|. b5) . b4) * (b3 . b5)) + (((b3 .|. b6) . b4) * (b3 . b6))) + (((b3 .|. b7) . b4) * (b3 . b7));
:: PROB_2:th 46
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5 being Event of b2
st 0 < b3 . b5
holds (b3 .|. b5) . b4 = b3 . b4
iff
b4,b5 are_independent_respect_to b3;
:: PROB_2:th 47
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5 being Event of b2
st 0 < b3 . b5 &
b3 . b5 < 1 &
(b3 .|. b5) . b4 = (b3 .|. (([#] b2) \ b5)) . b4
holds b4,b5 are_independent_respect_to b3;
:: PROB_2:th 48
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Probability of b2
for b4, b5 being Event of b2
st 0 < b3 . b5
holds (((b3 . b4) + (b3 . b5)) - 1) / (b3 . b5) <= (b3 .|. b5) . b4;
:: PROB_2:th 49
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Event of b2
for b5 being Probability of b2
st 0 < b5 . b3 & 0 < b5 . b4
holds (b5 .|. b4) . b3 = (((b5 .|. b3) . b4) * (b5 . b3)) / (b5 . b4);
:: PROB_2:th 50
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4, b5 being Event of b2
for b6 being Probability of b2
st 0 < b6 . b3 & b5 = b4 ` & 0 < b6 . b4 & 0 < b6 . b5
holds (b6 .|. b3) . b4 = (((b6 .|. b4) . b3) * (b6 . b4)) / ((((b6 .|. b4) . b3) * (b6 . b4)) + (((b6 .|. b5) . b3) * (b6 . b5))) &
(b6 .|. b3) . b5 = (((b6 .|. b5) . b3) * (b6 . b5)) / ((((b6 .|. b4) . b3) * (b6 . b4)) + (((b6 .|. b5) . b3) * (b6 . b5)));
:: PROB_2:th 51
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4, b5, b6 being Event of b2
for b7 being Probability of b2
st 0 < b7 . b3 & 0 < b7 . b4 & 0 < b7 . b5 & 0 < b7 . b6 & b4 misses b5 & b6 = (b4 \/ b5) `
holds (b7 .|. b3) . b4 = (((b7 .|. b4) . b3) * (b7 . b4)) / (((((b7 .|. b4) . b3) * (b7 . b4)) + (((b7 .|. b5) . b3) * (b7 . b5))) + (((b7 .|. b6) . b3) * (b7 . b6))) &
(b7 .|. b3) . b5 = (((b7 .|. b5) . b3) * (b7 . b5)) / (((((b7 .|. b4) . b3) * (b7 . b4)) + (((b7 .|. b5) . b3) * (b7 . b5))) + (((b7 .|. b6) . b3) * (b7 . b6))) &
(b7 .|. b3) . b6 = (((b7 .|. b6) . b3) * (b7 . b6)) / (((((b7 .|. b4) . b3) * (b7 . b4)) + (((b7 .|. b5) . b3) * (b7 . b5))) + (((b7 .|. b6) . b3) * (b7 . b6)));
:: PROB_2:th 52
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Event of b2
for b5 being Probability of b2
st 0 < b5 . b4
holds 1 - ((b5 . (([#] b2) \ b3)) / (b5 . b4)) <= (b5 .|. b4) . b3;