Article PROB_4, MML version 4.99.1005
:: PROB_4:funcnot 1 => PROB_4:func 1
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;
let a4 be Element of NAT;
redefine func a3 . a4 -> Element of a2;
end;
:: PROB_4:th 1
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
rng b3 c= b2;
:: PROB_4:th 2
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Relation-like Function-like set holds
b3 is SetSequence of b2
iff
b3 is Function-like quasi_total Relation of NAT,b2;
:: PROB_4:sch 1
scheme PROB_4:sch 1
{F1 -> set,
F2 -> non empty compl-closed sigma-multiplicative Element of bool bool F1(),
F3 -> Element of F2()}:
ex b1 being SetSequence of F2() st
for b2 being Element of NAT holds
b1 . b2 = F3(b2)
:: PROB_4:exreg 1
registration
let a1 be set;
cluster Relation-like Function-like non empty total quasi_total disjoint_valued Relation of NAT,bool a1;
end;
:: PROB_4:exreg 2
registration
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
cluster Relation-like Function-like non empty total quasi_total disjoint_valued SetSequence of a2;
end;
:: PROB_4:th 3
theorem
for b1 being set
for b2, b3 being Element of bool b1 holds
ex b4 being Function-like quasi_total Relation of NAT,bool b1 st
b4 . 0 = b2 &
b4 . 1 = b3 &
(for b5 being Element of NAT
st 1 < b5
holds b4 . b5 = {});
:: PROB_4:th 4
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3, b4 being Element of bool b1
st b3 misses b4 &
b2 . 0 = b3 &
b2 . 1 = b4 &
(for b5 being Element of NAT
st 1 < b5
holds b2 . b5 = {})
holds b2 is disjoint_valued & Union b2 = b3 \/ b4;
:: PROB_4:th 5
theorem
for b1 being set
for b2 being non empty set holds
b2 is non empty compl-closed sigma-multiplicative Element of bool bool b1
iff
b2 c= bool b1 &
(for b3 being Function-like quasi_total Relation of NAT,bool b1
st for b4 being Element of NAT holds
b3 . b4 in b2
holds Union b3 in b2) &
(for b3 being Element of bool b1
st b3 in b2
holds b3 ` in b2);
:: PROB_4: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 Probability of b2
for b4, b5 being Event of b2 holds
b3 . (b4 \ b5) = (b3 . (b4 \/ b5)) - (b3 . b5);
:: PROB_4: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 Probability of b2
for b4, b5 being Event of b2
st b4 c= b5 & b3 . b5 = 0
holds b3 . b4 = 0;
:: PROB_4:th 8
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
for b5 being Element of NAT holds
b4 . (b3 . b5) = 0
iff
b4 . Union b3 = 0;
:: PROB_4: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 Probability of b2 holds
for b5 being set
st b5 in rng b3
holds b4 . b5 = 0
iff
b4 . union rng b3 = 0;
:: PROB_4:th 10
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 = b2
holds Partial_Sums b1 = Ser b2;
:: PROB_4:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 = b2 & b1 is bounded_above
holds sup b1 = sup rng b2;
:: PROB_4:th 12
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 = b2 & b1 is bounded_below
holds inf b1 = inf rng b2;
:: PROB_4:th 13
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 = b2 & b1 is nonnegative-yielding & b1 is summable
holds Sum b1 = SUM b2;
:: PROB_4:th 14
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 holds
b3 is sigma_Measure of b2;
:: PROB_4:funcnot 2 => PROB_4: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 Probability of a2;
func P2M A3 -> sigma_Measure of a2 equals
a3;
end;
:: PROB_4: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 Probability of b2 holds
P2M b3 = b3;
:: PROB_4:th 15
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
st b3 . b1 = R_EAL 1
holds b3 is Probability of b2;
:: PROB_4:funcnot 3 => PROB_4: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 sigma_Measure of a2;
assume a3 . a1 = R_EAL 1;
func M2P A3 -> Probability of a2 equals
a3;
end;
:: PROB_4: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 sigma_Measure of b2
st b3 . b1 = R_EAL 1
holds M2P b3 = b3;
:: PROB_4:th 16
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is non-decreasing(b1)
holds Partial_Union b2 = b2;
:: PROB_4:th 17
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is non-decreasing(b1)
holds (Partial_Diff_Union b2) . 0 = b2 . 0 &
(for b3 being Element of NAT holds
(Partial_Diff_Union b2) . (b3 + 1) = (b2 . (b3 + 1)) \ (b2 . b3));
:: PROB_4:th 18
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is non-decreasing(b1)
for b3 being Element of NAT holds
b2 . (b3 + 1) = ((Partial_Diff_Union b2) . (b3 + 1)) \/ (b2 . b3);
:: PROB_4:th 19
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is non-decreasing(b1)
for b3 being Element of NAT holds
(Partial_Diff_Union b2) . (b3 + 1) misses b2 . b3;
:: PROB_4:th 20
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
st b3 is non-decreasing(b1)
holds @Partial_Union b3 = b3;
:: PROB_4:th 21
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
st b3 is non-decreasing(b1)
holds (@Partial_Diff_Union b3) . 0 = b3 . 0 &
(for b4 being Element of NAT holds
(@Partial_Diff_Union b3) . (b4 + 1) = (b3 . (b4 + 1)) \ (b3 . b4));
:: PROB_4:th 22
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
st b3 is non-decreasing(b1)
for b4 being Element of NAT holds
(@Partial_Diff_Union b3) . (b4 + 1) misses b3 . b4;
:: PROB_4:prednot 1 => PROB_4: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;
pred A3 is_complete A2 means
for b1 being Element of bool a1
for b2 being set
st b2 in a2 & b1 c= b2 & a3 . b2 = 0
holds b1 in a2;
end;
:: PROB_4:dfs 3
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;
To prove
a3 is_complete a2
it is sufficient to prove
thus for b1 being Element of bool a1
for b2 being set
st b2 in a2 & b1 c= b2 & a3 . b2 = 0
holds b1 in a2;
:: PROB_4:def 3
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 holds
b3 is_complete b2
iff
for b4 being Element of bool b1
for b5 being set
st b5 in b2 & b4 c= b5 & b3 . b5 = 0
holds b4 in b2;
:: PROB_4: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 holds
b3 is_complete b2
iff
P2M b3 is_complete b2;
:: PROB_4:modenot 1 => PROB_4:mode 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;
mode thin of A3 -> Element of bool a1 means
ex b1 being set st
b1 in a2 & it c= b1 & a3 . b1 = 0;
end;
:: PROB_4: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 Probability of a2;
let a4 be Element of bool a1;
To prove
a4 is thin of a3
it is sufficient to prove
thus ex b1 being set st
b1 in a2 & a4 c= b1 & a3 . b1 = 0;
:: PROB_4: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 Probability of b2
for b4 being Element of bool b1 holds
b4 is thin of b3
iff
ex b5 being set st
b5 in b2 & b4 c= b5 & b3 . b5 = 0;
:: PROB_4: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 being Element of bool b1 holds
b4 is thin of b3
iff
b4 is thin of P2M b3;
:: PROB_4: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 holds
{} is thin of b3;
:: PROB_4: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, b5 being set
st b4 in b2 & b5 in b2
for b6, b7 being thin of b3
st b4 \/ b6 = b5 \/ b7
holds b3 . b4 = b3 . b5;
:: PROB_4:funcnot 4 => PROB_4: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;
func COM(A2,A3) -> non empty Element of bool bool a1 means
for b1 being set holds
b1 in it
iff
ex b2 being set st
b2 in a2 &
(ex b3 being thin of a3 st
b1 = b2 \/ b3);
end;
:: PROB_4: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 being non empty Element of bool bool b1 holds
b4 = COM(b2,b3)
iff
for b5 being set holds
b5 in b4
iff
ex b6 being set st
b6 in b2 &
(ex b7 being thin of b3 st
b5 = b6 \/ b7);
:: PROB_4: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 thin of b3 holds
b4 in COM(b2,b3);
:: PROB_4: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 holds
COM(b2,b3) = COM(b2,P2M b3);
:: PROB_4:funcnot 5 => PROB_4:func 5
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 Element of COM(a2,a3);
func P_COM2M_COM A4 -> Element of COM(a2,P2M a3) equals
a4;
end;
:: PROB_4: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 being Element of COM(b2,b3) holds
P_COM2M_COM b4 = b4;
:: PROB_4:th 29
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 holds
b2 c= COM(b2,b3);
:: PROB_4:funcnot 6 => PROB_4:func 6
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 Element of COM(a2,a3);
func ProbPart A4 -> non empty Element of bool bool a1 means
for b1 being set holds
b1 in it
iff
b1 in a2 & b1 c= a4 & a4 \ b1 is thin of a3;
end;
:: PROB_4: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 Element of COM(b2,b3)
for b5 being non empty Element of bool bool b1 holds
b5 = ProbPart b4
iff
for b6 being set holds
b6 in b5
iff
b6 in b2 & b6 c= b4 & b4 \ b6 is thin of b3;
:: PROB_4:th 30
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 Element of COM(b2,b3) holds
ProbPart b4 = MeasPart P_COM2M_COM b4;
:: PROB_4: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 being Element of COM(b2,b3)
for b5, b6 being set
st b5 in ProbPart b4 & b6 in ProbPart b4
holds b3 . b5 = b3 . b6;
:: PROB_4: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 being Function-like quasi_total Relation of NAT,COM(b2,b3) holds
ex b5 being SetSequence of b2 st
for b6 being Element of NAT holds
b5 . b6 in ProbPart (b4 . b6);
:: PROB_4: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 being Function-like quasi_total Relation of NAT,COM(b2,b3)
for b5 being SetSequence of b2 holds
ex b6 being Function-like quasi_total Relation of NAT,bool b1 st
for b7 being Element of NAT holds
b6 . b7 = (b4 . b7) \ (b5 . b7);
:: PROB_4: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 being Function-like quasi_total Relation of NAT,bool b1
st for b5 being Element of NAT holds
b4 . b5 is thin of b3
holds ex b5 being SetSequence of b2 st
for b6 being Element of NAT holds
b4 . b6 c= b5 . b6 & b3 . (b5 . b6) = 0;
:: PROB_4: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 being non empty Element of bool bool b1
st for b5 being set holds
b5 in b4
iff
ex b6 being set st
b6 in b2 &
(ex b7 being thin of b3 st
b5 = b6 \/ b7)
holds b4 is non empty compl-closed sigma-multiplicative Element of bool bool b1;
:: PROB_4:funcnot 7 => PROB_4:func 7
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;
redefine func COM(a2,a3) -> non empty compl-closed sigma-multiplicative Element of bool bool a1;
end;
:: PROB_4:modenot 2 => PROB_4:mode 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;
redefine mode thin of a3 -> Event of COM(a2,a3);
end;
:: PROB_4: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 set holds
b4 in COM(b2,b3)
iff
ex b5, b6 being set st
b5 in b2 & b6 in b2 & b5 c= b4 & b4 c= b6 & b3 . (b6 \ b5) = 0;
:: PROB_4: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 being Probability of b2
for b4 being non empty Element of bool bool b1
st for b5 being set holds
b5 in b4
iff
ex b6, b7 being set st
b6 in b2 & b7 in b2 & b6 c= b5 & b5 c= b7 & b3 . (b7 \ b6) = 0
holds b4 = COM(b2,b3);
:: PROB_4:funcnot 8 => PROB_4:func 8
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;
func COM A3 -> Probability of COM(a2,a3) means
for b1 being set
st b1 in a2
for b2 being thin of a3 holds
it . (b1 \/ b2) = a3 . b1;
end;
:: PROB_4:def 8
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 Probability of COM(b2,b3) holds
b4 = COM b3
iff
for b5 being set
st b5 in b2
for b6 being thin of b3 holds
b4 . (b5 \/ b6) = b3 . b5;
:: PROB_4: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 holds
COM b3 = COM P2M b3;
:: PROB_4: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 being Probability of b2 holds
COM b3 is_complete COM(b2,b3);
:: PROB_4: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 being Event of b2 holds
b3 . b4 = (COM b3) . b4;
:: PROB_4:th 41
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 thin of b3 holds
(COM b3) . b4 = 0;
:: PROB_4: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 being Element of COM(b2,b3)
for b5 being set
st b5 in ProbPart b4
holds b3 . b5 = (COM b3) . b4;