Article PROB_3, MML version 4.99.1005
:: PROB_3:th 1
theorem
for b1 being Relation-like Function-like FinSequence-like set holds
not 0 in proj1 b1;
:: PROB_3:th 2
theorem
for b1 being natural set
for b2 being Relation-like Function-like FinSequence-like set holds
b1 in proj1 b2
iff
b1 <> 0 & b1 <= len b2;
:: PROB_3:th 3
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st ex b3 being natural set st
for b4 being natural set
st b3 <= b4
holds b2 . b4 = b1
holds b2 is convergent & lim b2 = b1;
:: PROB_3:th 4
theorem
for b1 being natural set
for b2 being non empty set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4 being SetSequence of b3
for b5 being Probability of b3 holds
0 <= (b5 * b4) . b1;
:: PROB_3:th 5
theorem
for b1 being natural set
for b2 being non empty set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4, b5 being SetSequence of b3
for b6 being Probability of b3
st b4 . b1 c= b5 . b1
holds (b6 * b4) . b1 <= (b6 * b5) . b1;
:: PROB_3:th 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 SetSequence of b2
for b4 being Probability of b2
st b3 is non-decreasing(b1)
holds b4 * b3 is non-decreasing;
:: PROB_3:th 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 SetSequence of b2
for b4 being Probability of b2
st b3 is non-increasing(b1)
holds b4 * b3 is non-increasing;
:: PROB_3:th 8
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
ex b3 being Function-like quasi_total Relation of NAT,bool b1 st
b3 . 0 = b2 . 0 &
(for b4 being natural set holds
b3 . (b4 + 1) = (b3 . b4) /\ (b2 . (b4 + 1)));
:: PROB_3:th 9
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
ex b3 being Function-like quasi_total Relation of NAT,bool b1 st
b3 . 0 = b2 . 0 &
(for b4 being natural set holds
b3 . (b4 + 1) = (b3 . b4) \/ (b2 . (b4 + 1)));
:: PROB_3:funcnot 1 => PROB_3:func 1
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
func Partial_Intersection A2 -> Function-like quasi_total Relation of NAT,bool a1 means
it . 0 = a2 . 0 &
(for b1 being natural set holds
it . (b1 + 1) = (it . b1) /\ (a2 . (b1 + 1)));
end;
:: PROB_3:def 1
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
b3 = Partial_Intersection b2
iff
b3 . 0 = b2 . 0 &
(for b4 being natural set holds
b3 . (b4 + 1) = (b3 . b4) /\ (b2 . (b4 + 1)));
:: PROB_3:funcnot 2 => PROB_3:func 2
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
func Partial_Union A2 -> Function-like quasi_total Relation of NAT,bool a1 means
it . 0 = a2 . 0 &
(for b1 being natural set holds
it . (b1 + 1) = (it . b1) \/ (a2 . (b1 + 1)));
end;
:: PROB_3:def 2
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
b3 = Partial_Union b2
iff
b3 . 0 = b2 . 0 &
(for b4 being natural set holds
b3 . (b4 + 1) = (b3 . b4) \/ (b2 . (b4 + 1)));
:: PROB_3:th 10
theorem
for b1 being natural set
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2 holds
(Partial_Intersection b3) . b1 c= b3 . b1;
:: PROB_3:th 11
theorem
for b1 being natural set
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2 holds
b3 . b1 c= (Partial_Union b3) . b1;
:: PROB_3:th 12
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
Partial_Intersection b2 is non-increasing(b1);
:: PROB_3:th 13
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
Partial_Union b2 is non-decreasing(b1);
:: PROB_3:th 14
theorem
for b1 being natural set
for b2, b3 being set
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
b3 in (Partial_Intersection b4) . b1
iff
for b5 being natural set
st b5 <= b1
holds b3 in b4 . b5;
:: PROB_3:th 15
theorem
for b1 being natural set
for b2, b3 being set
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
b3 in (Partial_Union b4) . b1
iff
ex b5 being natural set st
b5 <= b1 & b3 in b4 . b5;
:: PROB_3:th 16
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
Intersection Partial_Intersection b2 = Intersection b2;
:: PROB_3:th 17
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
Union Partial_Union b2 = Union b2;
:: PROB_3:th 18
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
ex b3 being Function-like quasi_total Relation of NAT,bool b1 st
b3 . 0 = b2 . 0 &
(for b4 being natural set holds
b3 . (b4 + 1) = (b2 . (b4 + 1)) \ ((Partial_Union b2) . b4));
:: PROB_3:funcnot 3 => PROB_3:func 3
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
func Partial_Diff_Union A2 -> Function-like quasi_total Relation of NAT,bool a1 means
it . 0 = a2 . 0 &
(for b1 being natural set holds
it . (b1 + 1) = (a2 . (b1 + 1)) \ ((Partial_Union a2) . b1));
end;
:: PROB_3:def 3
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
b3 = Partial_Diff_Union b2
iff
b3 . 0 = b2 . 0 &
(for b4 being natural set holds
b3 . (b4 + 1) = (b2 . (b4 + 1)) \ ((Partial_Union b2) . b4));
:: PROB_3:th 19
theorem
for b1 being natural set
for b2, b3 being set
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
b3 in (Partial_Diff_Union b4) . b1
iff
b3 in b4 . b1 &
(for b5 being natural set
st b5 < b1
holds not b3 in b4 . b5);
:: PROB_3:th 20
theorem
for b1 being natural set
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2 holds
(Partial_Diff_Union b3) . b1 c= b3 . b1;
:: PROB_3:th 21
theorem
for b1 being natural set
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2 holds
(Partial_Diff_Union b3) . b1 c= (Partial_Union b3) . b1;
:: PROB_3:th 22
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
Partial_Union Partial_Diff_Union b2 = Partial_Union b2;
:: PROB_3:th 23
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
Union Partial_Diff_Union b2 = Union b2;
:: PROB_3:attrnot 1 => PROB_3:attr 1
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
redefine attr a2 is disjoint_valued means
for b1, b2 being natural set
st b1 <> b2
holds a2 . b1 misses a2 . b2;
end;
:: PROB_3:dfs 4
definiens
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
To prove
a1 is disjoint_valued
it is sufficient to prove
thus for b1, b2 being natural set
st b1 <> b2
holds a2 . b1 misses a2 . b2;
:: PROB_3:def 4
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 is disjoint_valued
iff
for b3, b4 being natural set
st b3 <> b4
holds b2 . b3 misses b2 . b4;
:: PROB_3:th 24
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
Partial_Diff_Union b2 is disjoint_valued;
:: PROB_3:funcnot 4 => PROB_3:func 4
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
func @Partial_Intersection A3 -> SetSequence of a2 equals
Partial_Intersection a3;
end;
:: PROB_3:def 5
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
@Partial_Intersection b3 = Partial_Intersection b3;
:: PROB_3:funcnot 5 => PROB_3:func 5
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
func @Partial_Union A3 -> SetSequence of a2 equals
Partial_Union a3;
end;
:: PROB_3:def 6
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
@Partial_Union b3 = Partial_Union b3;
:: PROB_3:funcnot 6 => PROB_3:func 6
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
func @Partial_Diff_Union A3 -> SetSequence of a2 equals
Partial_Diff_Union a3;
end;
:: PROB_3:def 7
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
@Partial_Diff_Union b3 = Partial_Diff_Union b3;
:: PROB_3:th 25
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being SetSequence of b2
st b3 = @Partial_Intersection b4
holds b3 . 0 = b4 . 0 &
(for b5 being natural set holds
b3 . (b5 + 1) = (b3 . b5) /\ (b4 . (b5 + 1)));
:: PROB_3:th 26
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being SetSequence of b2
st b3 = @Partial_Union b4
holds b3 . 0 = b4 . 0 &
(for b5 being natural set holds
b3 . (b5 + 1) = (b3 . b5) \/ (b4 . (b5 + 1)));
:: PROB_3:th 27
theorem
for b1 being natural set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4 being SetSequence of b3 holds
(@Partial_Intersection b4) . b1 c= b4 . b1;
:: PROB_3:th 28
theorem
for b1 being natural set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4 being SetSequence of b3 holds
b4 . b1 c= (@Partial_Union b4) . b1;
:: PROB_3:th 29
theorem
for b1 being natural set
for b2, b3 being set
for b4 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b5 being SetSequence of b4 holds
b3 in (@Partial_Intersection b5) . b1
iff
for b6 being natural set
st b6 <= b1
holds b3 in b5 . b6;
:: PROB_3:th 30
theorem
for b1 being natural set
for b2, b3 being set
for b4 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b5 being SetSequence of b4 holds
b3 in (@Partial_Union b5) . b1
iff
ex b6 being natural set st
b6 <= b1 & b3 in b5 . b6;
:: PROB_3:th 31
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
@Partial_Intersection b3 is non-increasing(b1);
:: PROB_3:th 32
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
@Partial_Union b3 is non-decreasing(b1);
:: PROB_3:th 33
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
Intersection @Partial_Intersection b3 = Intersection b3;
:: PROB_3:th 34
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
Union @Partial_Union b3 = Union b3;
:: PROB_3:th 35
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being SetSequence of b2
st b3 = @Partial_Diff_Union b4
holds b3 . 0 = b4 . 0 &
(for b5 being natural set holds
b3 . (b5 + 1) = (b4 . (b5 + 1)) \ ((@Partial_Union b4) . b5));
:: PROB_3:th 36
theorem
for b1 being natural set
for b2, b3 being set
for b4 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b5 being SetSequence of b4 holds
b3 in (@Partial_Diff_Union b5) . b1
iff
b3 in b5 . b1 &
(for b6 being natural set
st b6 < b1
holds not b3 in b5 . b6);
:: PROB_3:th 37
theorem
for b1 being natural set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4 being SetSequence of b3 holds
(@Partial_Diff_Union b4) . b1 c= b4 . b1;
:: PROB_3:th 38
theorem
for b1 being natural set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4 being SetSequence of b3 holds
(@Partial_Diff_Union b4) . b1 c= (@Partial_Union b4) . b1;
:: PROB_3:th 39
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
@Partial_Union @Partial_Diff_Union b3 = @Partial_Union b3;
:: PROB_3:th 40
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
Union @Partial_Diff_Union b3 = Union b3;
:: PROB_3:th 41
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
@Partial_Diff_Union b3 is disjoint_valued;
:: PROB_3: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 SetSequence of b2
for b4 being Probability of b2 holds
b4 * @Partial_Union b3 is non-decreasing;
:: PROB_3: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 SetSequence of b2
for b4 being Probability of b2 holds
b4 * @Partial_Intersection b3 is non-increasing;
:: PROB_3: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 SetSequence of b2
for b4 being Probability of b2 holds
Partial_Sums (b4 * b3) is non-decreasing;
:: PROB_3: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 SetSequence of b2
for b4 being Probability of b2 holds
(b4 * @Partial_Union b3) . 0 = (Partial_Sums (b4 * b3)) . 0;
:: PROB_3: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 SetSequence of b2
for b4 being Probability of b2 holds
b4 * @Partial_Union b3 is convergent &
lim (b4 * @Partial_Union b3) = sup (b4 * @Partial_Union b3) &
lim (b4 * @Partial_Union b3) = b4 . Union b3;
:: PROB_3: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 SetSequence of b2
st b3 is disjoint_valued
for b4, b5 being natural set
st b4 < b5
holds (@Partial_Union b3) . b4 misses b3 . b5;
:: PROB_3:th 48
theorem
for b1 being natural set
for b2 being non empty set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4 being SetSequence of b3
for b5 being Probability of b3
st b4 is disjoint_valued
holds (b5 * @Partial_Union b4) . b1 = (Partial_Sums (b5 * b4)) . b1;
:: PROB_3: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 being SetSequence of b2
for b4 being Probability of b2
st b3 is disjoint_valued
holds b4 * @Partial_Union b3 = Partial_Sums (b4 * b3);
:: PROB_3: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 being SetSequence of b2
for b4 being Probability of b2
st b3 is disjoint_valued
holds Partial_Sums (b4 * b3) is convergent &
lim Partial_Sums (b4 * b3) = sup Partial_Sums (b4 * b3) &
lim Partial_Sums (b4 * b3) = b4 . Union b3;
:: PROB_3: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 being SetSequence of b2
for b4 being Probability of b2
st b3 is disjoint_valued
holds b4 . Union b3 = Sum (b4 * b3);
:: PROB_3:funcnot 7 => PROB_3:func 7
definition
let a1 be set;
let a2 be FinSequence of bool a1;
let a3 be natural set;
redefine func a2 . a3 -> Element of bool a1;
end;
:: PROB_3:th 52
theorem
for b1 being set holds
ex b2 being FinSequence of bool b1 st
for b3 being natural set
st b3 in dom b2
holds b2 . b3 = b1;
:: PROB_3:th 53
theorem
for b1 being set
for b2 being FinSequence of bool b1 holds
union rng b2 is Element of bool b1;
:: PROB_3:funcnot 8 => PROB_3:func 8
definition
let a1 be set;
let a2 be FinSequence of bool a1;
redefine func Union a2 -> Element of bool a1;
end;
:: PROB_3:th 54
theorem
for b1, b2 being set
for b3 being FinSequence of bool b1 holds
b2 in Union b3
iff
ex b4 being natural set st
b4 in dom b3 & b2 in b3 . b4;
:: PROB_3:funcnot 9 => PROB_3:func 9
definition
let a1 be set;
let a2 be FinSequence of bool a1;
func Complement A2 -> FinSequence of bool a1 means
len it = len a2 &
(for b1 being natural set
st b1 in dom it
holds it . b1 = (a2 . b1) `);
end;
:: PROB_3:def 8
theorem
for b1 being set
for b2, b3 being FinSequence of bool b1 holds
b3 = Complement b2
iff
len b3 = len b2 &
(for b4 being natural set
st b4 in dom b3
holds b3 . b4 = (b2 . b4) `);
:: PROB_3:funcnot 10 => PROB_3:func 10
definition
let a1 be set;
let a2 be FinSequence of bool a1;
func Intersection A2 -> Element of bool a1 equals
(Union Complement a2) `
if a2 <> {}
otherwise {};
end;
:: PROB_3:def 9
theorem
for b1 being set
for b2 being FinSequence of bool b1 holds
(b2 = {} or Intersection b2 = (Union Complement b2) `) &
(b2 = {} implies Intersection b2 = {});
:: PROB_3:th 55
theorem
for b1 being set
for b2 being FinSequence of bool b1 holds
dom Complement b2 = dom b2;
:: PROB_3:th 56
theorem
for b1, b2 being set
for b3 being FinSequence of bool b1
st b3 <> {}
holds b2 in Intersection b3
iff
for b4 being natural set
st b4 in dom b3
holds b2 in b3 . b4;
:: PROB_3:th 57
theorem
for b1, b2 being set
for b3 being FinSequence of bool b1
st b3 <> {}
holds b2 in meet rng b3
iff
for b4 being natural set
st b4 in dom b3
holds b2 in b3 . b4;
:: PROB_3:th 58
theorem
for b1 being set
for b2 being FinSequence of bool b1 holds
Intersection b2 = meet rng b2;
:: PROB_3:th 59
theorem
for b1 being set
for b2 being FinSequence of bool b1 holds
ex b3 being Function-like quasi_total Relation of NAT,bool b1 st
(for b4 being natural set
st b4 in dom b2
holds b3 . b4 = b2 . b4) &
(for b4 being natural set
st not b4 in dom b2
holds b3 . b4 = {});
:: PROB_3:th 60
theorem
for b1 being set
for b2 being FinSequence of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1
st (for b4 being natural set
st b4 in dom b2
holds b3 . b4 = b2 . b4) &
(for b4 being natural set
st not b4 in dom b2
holds b3 . b4 = {})
holds b3 . 0 = {} & Union b3 = Union b2;
:: PROB_3:modenot 1 => PROB_3:mode 1
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
mode FinSequence of A2 -> FinSequence of bool a1 means
for b1 being natural set
st b1 in dom it
holds it . b1 in a2;
end;
:: PROB_3:dfs 10
definiens
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be FinSequence of bool a1;
To prove
a3 is FinSequence of a2
it is sufficient to prove
thus for b1 being natural set
st b1 in dom a3
holds a3 . b1 in a2;
:: PROB_3:def 10
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being FinSequence of bool b1 holds
b3 is FinSequence of b2
iff
for b4 being natural set
st b4 in dom b3
holds b3 . b4 in b2;
:: PROB_3:funcnot 11 => PROB_3:func 11
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be FinSequence of a2;
let a4 be natural set;
redefine func a3 . a4 -> Event of a2;
end;
:: PROB_3:th 61
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being FinSequence of b2 holds
ex b4 being SetSequence of b2 st
(for b5 being natural set
st b5 in dom b3
holds b4 . b5 = b3 . b5) &
(for b5 being natural set
st not b5 in dom b3
holds b4 . b5 = {});
:: PROB_3:th 62
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being FinSequence of b2 holds
Union b3 in b2;
:: PROB_3:funcnot 12 => PROB_3:func 12
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be FinSequence of a2;
func @Complement A3 -> FinSequence of a2 equals
Complement a3;
end;
:: PROB_3:def 11
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being FinSequence of b2 holds
@Complement b3 = Complement b3;
:: PROB_3:th 63
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being FinSequence of b2 holds
Intersection b3 in b2;
:: PROB_3:th 64
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 FinSequence of b2 holds
dom (b3 * b4) = dom b4;
:: PROB_3:th 65
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 FinSequence of b2 holds
b3 * b4 is FinSequence of REAL;
:: PROB_3:funcnot 13 => PROB_3:func 13
definition
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be FinSequence of a2;
let a4 be Probability of a2;
redefine func a4 * a3 -> FinSequence of REAL;
end;
:: PROB_3:th 66
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 FinSequence of b2 holds
len (b3 * b4) = len b4;
:: PROB_3:th 67
theorem
for b1 being FinSequence of REAL
st len b1 = 0
holds Sum b1 = 0;
:: PROB_3:th 68
theorem
for b1 being FinSequence of REAL
st 1 <= len b1
holds ex b2 being Function-like quasi_total Relation of NAT,REAL st
b2 . 1 = b1 . 1 &
(for b3 being natural set
st 0 <> b3 & b3 < len b1
holds b2 . (b3 + 1) = (b2 . b3) + (b1 . (b3 + 1))) &
Sum b1 = b2 . len b1;
:: PROB_3:th 69
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 FinSequence of b2
for b5 being SetSequence of b2
st (for b6 being natural set
st b6 in dom b4
holds b5 . b6 = b4 . b6) &
(for b6 being natural set
st not b6 in dom b4
holds b5 . b6 = {})
holds Partial_Sums (b3 * b5) is convergent &
Sum (b3 * b5) = (Partial_Sums (b3 * b5)) . len b4 &
b3 . Union b5 <= Sum (b3 * b5) &
Sum (b3 * b4) = Sum (b3 * b5);
:: PROB_3:th 70
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 FinSequence of b2 holds
b3 . Union b4 <= Sum (b3 * b4) &
(b4 is disjoint_valued implies b3 . Union b4 = Sum (b3 * b4));
:: PROB_3:attrnot 2 => PROB_3:attr 2
definition
let a1 be set;
let a2 be Element of bool bool a1;
attr a2 is non-decreasing-closed means
for b1 being Function-like quasi_total Relation of NAT,bool a1
st b1 is non-decreasing(a1) &
(for b2 being natural set holds
b1 . b2 in a2)
holds Union b1 in a2;
end;
:: PROB_3:dfs 12
definiens
let a1 be set;
let a2 be Element of bool bool a1;
To prove
a2 is non-decreasing-closed
it is sufficient to prove
thus for b1 being Function-like quasi_total Relation of NAT,bool a1
st b1 is non-decreasing(a1) &
(for b2 being natural set holds
b1 . b2 in a2)
holds Union b1 in a2;
:: PROB_3:def 12
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 is non-decreasing-closed(b1)
iff
for b3 being Function-like quasi_total Relation of NAT,bool b1
st b3 is non-decreasing(b1) &
(for b4 being natural set holds
b3 . b4 in b2)
holds Union b3 in b2;
:: PROB_3:attrnot 3 => PROB_3:attr 3
definition
let a1 be set;
let a2 be Element of bool bool a1;
attr a2 is non-increasing-closed means
for b1 being Function-like quasi_total Relation of NAT,bool a1
st b1 is non-increasing(a1) &
(for b2 being natural set holds
b1 . b2 in a2)
holds Intersection b1 in a2;
end;
:: PROB_3:dfs 13
definiens
let a1 be set;
let a2 be Element of bool bool a1;
To prove
a2 is non-increasing-closed
it is sufficient to prove
thus for b1 being Function-like quasi_total Relation of NAT,bool a1
st b1 is non-increasing(a1) &
(for b2 being natural set holds
b1 . b2 in a2)
holds Intersection b1 in a2;
:: PROB_3:def 13
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 is non-increasing-closed(b1)
iff
for b3 being Function-like quasi_total Relation of NAT,bool b1
st b3 is non-increasing(b1) &
(for b4 being natural set holds
b3 . b4 in b2)
holds Intersection b3 in b2;
:: PROB_3:th 71
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 is non-decreasing-closed(b1)
iff
for b3 being Function-like quasi_total Relation of NAT,bool b1
st b3 is non-decreasing(b1) &
(for b4 being natural set holds
b3 . b4 in b2)
holds lim b3 in b2;
:: PROB_3:th 72
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 is non-increasing-closed(b1)
iff
for b3 being Function-like quasi_total Relation of NAT,bool b1
st b3 is non-increasing(b1) &
(for b4 being natural set holds
b3 . b4 in b2)
holds lim b3 in b2;
:: PROB_3:th 73
theorem
for b1 being set holds
bool b1 is non-decreasing-closed(b1) & bool b1 is non-increasing-closed(b1);
:: PROB_3:modenot 2 => PROB_3:mode 2
definition
let a1 be set;
mode MonotoneClass of A1 -> Element of bool bool a1 means
it is non-decreasing-closed(a1) & it is non-increasing-closed(a1);
end;
:: PROB_3:dfs 14
definiens
let a1 be set;
let a2 be Element of bool bool a1;
To prove
a2 is MonotoneClass of a1
it is sufficient to prove
thus a2 is non-decreasing-closed(a1) & a2 is non-increasing-closed(a1);
:: PROB_3:def 14
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 is MonotoneClass of b1
iff
b2 is non-decreasing-closed(b1) & b2 is non-increasing-closed(b1);
:: PROB_3:th 74
theorem
for b1, b2 being set holds
b1 is MonotoneClass of b2
iff
b1 c= bool b2 &
(for b3 being Function-like quasi_total Relation of NAT,bool b2
st b3 is monotone(b2) &
(for b4 being natural set holds
b3 . b4 in b1)
holds lim b3 in b1);
:: PROB_3:th 75
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1 holds
b2 is non empty compl-closed sigma-multiplicative Element of bool bool b1
iff
b2 is MonotoneClass of b1;
:: PROB_3:th 76
theorem
for b1 being non empty set holds
bool b1 is MonotoneClass of b1;
:: PROB_3:th 77
theorem
for b1 being non empty set
for b2 being Element of bool bool b1 holds
ex b3 being MonotoneClass of b1 st
b2 c= b3 &
(for b4 being set
st b2 c= b4 & b4 is MonotoneClass of b1
holds b3 c= b4);
:: PROB_3:funcnot 14 => PROB_3:func 14
definition
let a1 be non empty set;
let a2 be Element of bool bool a1;
func monotoneclass A2 -> MonotoneClass of a1 means
a2 c= it &
(for b1 being set
st a2 c= b1 & b1 is MonotoneClass of a1
holds it c= b1);
end;
:: PROB_3:def 15
theorem
for b1 being non empty set
for b2 being Element of bool bool b1
for b3 being MonotoneClass of b1 holds
b3 = monotoneclass b2
iff
b2 c= b3 &
(for b4 being set
st b2 c= b4 & b4 is MonotoneClass of b1
holds b3 c= b4);
:: PROB_3:th 78
theorem
for b1 being non empty set
for b2 being non empty cap-closed compl-closed Element of bool bool b1 holds
monotoneclass b2 is non empty cap-closed compl-closed Element of bool bool b1;
:: PROB_3:th 79
theorem
for b1 being non empty set
for b2 being non empty cap-closed compl-closed Element of bool bool b1 holds
sigma b2 = monotoneclass b2;