Article MATRPROB, MML version 4.99.1005
:: MATRPROB:funcnot 1 => MATRPROB:func 1
definition
let a1 be set;
let a2 be FinSequence of a1 *;
let a3 be natural set;
redefine func a2 . a3 -> FinSequence of a1;
end;
:: MATRPROB:funcnot 2 => MATRPROB:func 2
definition
let a1 be real set;
redefine func <*a1*> -> FinSequence of REAL;
end;
:: MATRPROB:th 1
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being non empty natural set
for b4 being FinSequence of b1 holds
len b4 = b3 &
(for b5 being natural set
st b5 in dom b4
holds b4 . b5 = b2)
iff
b4 = b3 |-> b2;
:: MATRPROB:th 2
theorem
for b1 being non empty set
for b2 being Element of NAT
for b3 being natural set
for b4, b5 being Element of b1 holds
ex b6 being FinSequence of b1 st
len b6 = b3 &
(for b7 being natural set
st b7 in Seg b3
holds (b7 in Seg b2 implies b6 . b7 = b4) & (b7 in Seg b2 or b6 . b7 = b5));
:: MATRPROB:th 3
theorem
for b1 being FinSequence of REAL
st for b2 being natural set
st b2 in dom b1
holds 0 <= b1 . b2
for 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)))
for b3, b4 being natural set
st b3 in dom b1 & b4 in dom b1 & b3 <= b4
holds b2 . b3 <= b2 . b4;
:: MATRPROB:th 4
theorem
for b1 being FinSequence of REAL
st 1 <= len b1 &
(for b2 being natural set
st b2 in dom b1
holds 0 <= b1 . b2)
for 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)))
for b3 being natural set
st b3 in dom b1
holds b1 . b3 <= b2 . b3;
:: MATRPROB:th 5
theorem
for b1 being FinSequence of REAL
st for b2 being natural set
st b2 in dom b1
holds 0 <= b1 . b2
for b2 being natural set
st b2 in dom b1
holds b1 . b2 <= Sum b1;
:: MATRPROB:th 6
theorem
for b1, b2 being Element of REAL
for b3 being natural set
for b4 being Function-like quasi_total Relation of NAT,REAL holds
ex b5 being Function-like quasi_total Relation of NAT,REAL st
b5 . 0 = b1 &
(for b6 being natural set holds
(b6 <> 0 & b6 <= b3 implies b5 . b6 = b4 . b6) &
(b6 <= b3 or b5 . b6 = b2));
:: MATRPROB:th 7
theorem
for b1 being FinSequence of REAL holds
ex b2 being Function-like quasi_total Relation of NAT,REAL st
b2 . 0 = 0 &
(for b3 being natural set
st b3 < len b1
holds b2 . (b3 + 1) = (b2 . b3) + (b1 . (b3 + 1))) &
Sum b1 = b2 . len b1;
:: MATRPROB:th 8
theorem
for b1 being natural set
for b2 being set
for b3 being FinSequence of b2 holds
b1 |-> b3 is FinSequence of b2 *;
:: MATRPROB:th 9
theorem
for b1 being Element of NAT
for b2 being natural set
for b3 being set
for b4, b5 being FinSequence of b3 holds
ex b6 being FinSequence of b3 * st
len b6 = b2 &
(for b7 being natural set
st b7 in Seg b2
holds (b7 in Seg b1 implies b6 . b7 = b4) & (b7 in Seg b1 or b6 . b7 = b5));
:: MATRPROB:th 10
theorem
for b1 being set
for b2 being Relation-like Function-like FinSequence-like set holds
b2 is tabular FinSequence of b1 *
iff
ex b3 being natural set st
for b4 being Element of NAT
st b4 in dom b2
holds ex b5 being FinSequence of b1 st
b2 . b4 = b5 & len b5 = b3;
:: MATRPROB:th 11
theorem
for b1 being set
for b2 being FinSequence of b1 * holds
ex b3 being natural set st
for b4 being Element of NAT
st b4 in dom b2
holds len (b2 . b4) = b3
iff
b2 is tabular FinSequence of b1 *;
:: MATRPROB:th 12
theorem
for b1, b2 being Element of NAT
for b3 being Relation-like Function-like FinSequence-like tabular set holds
[b1,b2] in Indices b3
iff
b1 in Seg len b3 & b2 in Seg width b3;
:: MATRPROB:th 13
theorem
for b1, b2 being Element of NAT
for b3 being non empty set
for b4 being tabular FinSequence of b3 * holds
[b1,b2] in Indices b4
iff
b1 in dom b4 & b2 in dom (b4 . b1);
:: MATRPROB:th 14
theorem
for b1, b2 being Element of NAT
for b3 being non empty set
for b4 being tabular FinSequence of b3 *
st [b1,b2] in Indices b4
holds b4 *(b1,b2) = (b4 . b1) . b2;
:: MATRPROB:th 15
theorem
for b1, b2 being Element of NAT
for b3 being non empty set
for b4 being tabular FinSequence of b3 * holds
[b1,b2] in Indices b4
iff
b1 in dom Col(b4,b2) & b2 in dom Line(b4,b1);
:: MATRPROB:th 16
theorem
for b1, b2 being non empty set
for b3 being tabular FinSequence of b1 *
for b4 being tabular FinSequence of b2 *
st b3 = b4
for b5 being Element of NAT
st b5 in dom b3
holds Line(b3,b5) = Line(b4,b5);
:: MATRPROB:th 17
theorem
for b1, b2 being non empty set
for b3 being tabular FinSequence of b1 *
for b4 being tabular FinSequence of b2 *
st b3 = b4
for b5 being Element of NAT
st b5 in Seg width b3
holds Col(b3,b5) = Col(b4,b5);
:: MATRPROB:th 18
theorem
for b1 being non empty set
for b2, b3 being natural set
for b4 being FinSequence of b1
st len b4 = b2
holds b3 |-> b4 is Matrix of b3,b2,b1;
:: MATRPROB:th 19
theorem
for b1 being non empty set
for b2 being Element of NAT
for b3, b4 being natural set
for b5, b6 being FinSequence of b1
st len b5 = b3 & len b6 = b3
holds ex b7 being Matrix of b4,b3,b1 st
for b8 being natural set
st b8 in Seg b4
holds (b8 in Seg b2 implies b7 . b8 = b5) & (b8 in Seg b2 or b7 . b8 = b6);
:: MATRPROB:funcnot 3 => MATRPROB:func 3
definition
let a1 be FinSequence of REAL *;
func Sum A1 -> FinSequence of REAL means
len it = len a1 &
(for b1 being Element of NAT
st b1 in dom it
holds it . b1 = Sum (a1 . b1));
end;
:: MATRPROB:def 1
theorem
for b1 being FinSequence of REAL *
for b2 being FinSequence of REAL holds
b2 = Sum b1
iff
len b2 = len b1 &
(for b3 being Element of NAT
st b3 in dom b2
holds b2 . b3 = Sum (b1 . b3));
:: MATRPROB:funcnot 4 => MATRPROB:func 3
notation
let a1 be tabular FinSequence of REAL *;
synonym LineSum a1 for Sum a1;
end;
:: MATRPROB:th 20
theorem
for b1 being tabular FinSequence of REAL * holds
len Sum b1 = len b1 &
(for b2 being Element of NAT
st b2 in Seg len b1
holds (Sum b1) . b2 = Sum Line(b1,b2));
:: MATRPROB:funcnot 5 => MATRPROB:func 4
definition
let a1 be tabular FinSequence of REAL *;
func ColSum A1 -> FinSequence of REAL means
len it = width a1 &
(for b1 being natural set
st b1 in Seg width a1
holds it . b1 = Sum Col(a1,b1));
end;
:: MATRPROB:def 2
theorem
for b1 being tabular FinSequence of REAL *
for b2 being FinSequence of REAL holds
b2 = ColSum b1
iff
len b2 = width b1 &
(for b3 being natural set
st b3 in Seg width b1
holds b2 . b3 = Sum Col(b1,b3));
:: MATRPROB:th 21
theorem
for b1 being tabular FinSequence of REAL *
st 0 < width b1
holds Sum b1 = ColSum (b1 @);
:: MATRPROB:th 22
theorem
for b1 being tabular FinSequence of REAL * holds
ColSum b1 = Sum (b1 @);
:: MATRPROB:funcnot 6 => MATRPROB:func 5
definition
let a1 be tabular FinSequence of REAL *;
func SumAll A1 -> Element of REAL equals
Sum Sum a1;
end;
:: MATRPROB:def 3
theorem
for b1 being tabular FinSequence of REAL * holds
SumAll b1 = Sum Sum b1;
:: MATRPROB:th 23
theorem
for b1 being tabular FinSequence of REAL *
st len b1 = 0
holds SumAll b1 = 0;
:: MATRPROB:th 24
theorem
for b1 being natural set
for b2 being Matrix of b1,0,REAL holds
SumAll b2 = 0;
:: MATRPROB:th 25
theorem
for b1 being Element of NAT
for b2, b3 being natural set
for b4 being Matrix of b2,b1,REAL
for b5 being Matrix of b3,b1,REAL holds
Sum (b4 ^ b5) = (Sum b4) ^ Sum b5;
:: MATRPROB:th 26
theorem
for b1, b2 being tabular FinSequence of REAL * holds
(Sum b1) + Sum b2 = Sum (b1 ^^ b2);
:: MATRPROB:th 27
theorem
for b1, b2 being tabular FinSequence of REAL *
st len b1 = len b2
holds (SumAll b1) + SumAll b2 = SumAll (b1 ^^ b2);
:: MATRPROB:th 28
theorem
for b1 being tabular FinSequence of REAL * holds
SumAll b1 = SumAll (b1 @);
:: MATRPROB:th 29
theorem
for b1 being tabular FinSequence of REAL * holds
SumAll b1 = Sum ColSum b1;
:: MATRPROB:th 30
theorem
for b1, b2 being FinSequence of REAL
st len b1 = len b2
holds len mlt(b1,b2) = len b1;
:: MATRPROB:th 31
theorem
for b1 being Element of NAT
for b2 being Element of b1 -tuples_on REAL holds
mlt(b1 |-> 1,b2) = b2;
:: MATRPROB:th 32
theorem
for b1 being FinSequence of REAL holds
mlt((len b1) |-> 1,b1) = b1;
:: MATRPROB:th 33
theorem
for b1, b2 being FinSequence of REAL
st (for b3 being Element of NAT
st b3 in dom b1
holds 0 <= b1 . b3) &
(for b3 being Element of NAT
st b3 in dom b2
holds 0 <= b2 . b3)
for b3 being Element of NAT
st b3 in dom mlt(b1,b2)
holds 0 <= (mlt(b1,b2)) . b3;
:: MATRPROB:th 34
theorem
for b1 being Element of NAT
for b2, b3 being Element of b1 -tuples_on REAL
for b4, b5 being Element of b1 -tuples_on the carrier of F_Real
st b2 = b4 & b3 = b5
holds mlt(b2,b3) = mlt(b4,b5);
:: MATRPROB:th 35
theorem
for b1, b2 being FinSequence of REAL
for b3, b4 being FinSequence of the carrier of F_Real
st len b1 = len b2 & b1 = b3 & b2 = b4
holds mlt(b1,b2) = mlt(b3,b4);
:: MATRPROB:th 36
theorem
for b1 being FinSequence of REAL
for b2 being FinSequence of the carrier of F_Real
st b1 = b2
holds Sum b1 = Sum b2;
:: MATRPROB:funcnot 7 => EUCLID_2:func 1
notation
let a1, a2 be FinSequence of REAL;
synonym a1 "*" a2 for |(a1,a2)|;
end;
:: MATRPROB:th 37
theorem
for b1 being Element of NAT
for b2, b3 being Element of b1 -tuples_on REAL
for b4, b5 being Element of b1 -tuples_on the carrier of F_Real
st b2 = b4 & b3 = b5
holds |(b2,b3)| = b4 "*" b5;
:: MATRPROB:th 38
theorem
for b1, b2 being FinSequence of REAL
for b3, b4 being FinSequence of the carrier of F_Real
st len b1 = len b2 & b1 = b3 & b2 = b4
holds |(b1,b2)| = b3 "*" b4;
:: MATRPROB:th 39
theorem
for b1, b2, b3 being tabular FinSequence of REAL *
st width b2 = len b3
holds b1 = b2 * b3
iff
len b1 = len b2 &
width b1 = width b3 &
(for b4, b5 being Element of NAT
st [b4,b5] in Indices b1
holds b1 *(b4,b5) = |(Line(b2,b4),Col(b3,b5))|);
:: MATRPROB:th 40
theorem
for b1 being tabular FinSequence of REAL *
for b2 being FinSequence of REAL
st len b1 = len b2
for b3 being Element of NAT
st b3 in Seg len (b2 * b1)
holds (b2 * b1) . b3 = |(b2,Col(b1,b3))|;
:: MATRPROB:th 41
theorem
for b1 being tabular FinSequence of REAL *
for b2 being FinSequence of REAL
st width b1 = len b2 & 0 < width b1
for b3 being Element of NAT
st b3 in Seg len (b1 * b2)
holds (b1 * b2) . b3 = |(Line(b1,b3),b2)|;
:: MATRPROB:th 42
theorem
for b1, b2, b3 being tabular FinSequence of REAL *
st width b2 = len b3 & 0 < width b2 & 0 < width b3
holds b1 = b2 * b3
iff
len b1 = len b2 &
width b1 = width b3 &
(for b4 being Element of NAT
st b4 in Seg len b1
holds Line(b1,b4) = (Line(b2,b4)) * b3);
:: MATRPROB:funcreg 1
registration
let a1, a2, a3 be non empty natural set;
let a4 be Matrix of a1,a3,REAL;
let a5 be Matrix of a3,a2,REAL;
cluster a4 * a5 -> ;
end;
:: MATRPROB:funcnot 8 => MATRPROB:func 6
definition
let a1, a2 be FinSequence of REAL;
let a3 be tabular FinSequence of REAL *;
assume len a1 = len a3 & len a2 = width a3;
func QuadraticForm(A1,A3,A2) -> tabular FinSequence of REAL * means
len it = len a1 &
width it = len a2 &
(for b1, b2 being natural set
st [b1,b2] in Indices a3
holds it *(b1,b2) = ((a1 . b1) * (a3 *(b1,b2))) * (a2 . b2));
end;
:: MATRPROB:def 4
theorem
for b1, b2 being FinSequence of REAL
for b3 being tabular FinSequence of REAL *
st len b1 = len b3 & len b2 = width b3
for b4 being tabular FinSequence of REAL * holds
b4 = QuadraticForm(b1,b3,b2)
iff
len b4 = len b1 &
width b4 = len b2 &
(for b5, b6 being natural set
st [b5,b6] in Indices b3
holds b4 *(b5,b6) = ((b1 . b5) * (b3 *(b5,b6))) * (b2 . b6));
:: MATRPROB:th 43
theorem
for b1, b2 being FinSequence of REAL
for b3 being tabular FinSequence of REAL *
st len b1 = len b3 & len b2 = width b3 & 0 < len b1 & 0 < len b2
holds (QuadraticForm(b1,b3,b2)) @ = QuadraticForm(b2,b3 @,b1);
:: MATRPROB:th 44
theorem
for b1, b2 being FinSequence of REAL
for b3 being tabular FinSequence of REAL *
st len b1 = len b3 & len b2 = width b3 & 0 < len b1 & 0 < len b2
holds |(b1,b3 * b2)| = SumAll QuadraticForm(b1,b3,b2);
:: MATRPROB:th 45
theorem
for b1 being FinSequence of REAL holds
|(b1,(len b1) |-> 1)| = Sum b1;
:: MATRPROB:th 46
theorem
for b1, b2 being FinSequence of REAL
for b3 being tabular FinSequence of REAL *
st len b1 = len b3 & len b2 = width b3 & 0 < len b1 & 0 < len b2
holds |(b1 * b3,b2)| = SumAll QuadraticForm(b1,b3,b2);
:: MATRPROB:th 47
theorem
for b1, b2 being FinSequence of REAL
for b3 being tabular FinSequence of REAL *
st len b1 = len b3 & len b2 = width b3 & 0 < len b1 & 0 < len b2
holds |(b1 * b3,b2)| = |(b1,b3 * b2)|;
:: MATRPROB:th 48
theorem
for b1, b2 being FinSequence of REAL
for b3 being tabular FinSequence of REAL *
st len b2 = len b3 & len b1 = width b3 & 0 < len b1 & 0 < len b2
holds |(b3 * b1,b2)| = |(b1,b3 @ * b2)|;
:: MATRPROB:th 49
theorem
for b1, b2 being FinSequence of REAL
for b3 being tabular FinSequence of REAL *
st len b2 = len b3 & len b1 = width b3 & 0 < len b1 & 0 < len b2
holds |(b1,b2 * b3)| = |(b1 * (b3 @),b2)|;
:: MATRPROB:th 50
theorem
for b1 being FinSequence of REAL
for b2 being tabular FinSequence of REAL *
st len b1 = len b2 & b1 = (len b1) |-> 1
for b3 being Element of NAT
st b3 in Seg len (b1 * b2)
holds (b1 * b2) . b3 = Sum Col(b2,b3);
:: MATRPROB:th 51
theorem
for b1 being FinSequence of REAL
for b2 being tabular FinSequence of REAL *
st len b1 = width b2 & 0 < width b2 & b1 = (len b1) |-> 1
for b3 being Element of NAT
st b3 in Seg len (b2 * b1)
holds (b2 * b1) . b3 = Sum Line(b2,b3);
:: MATRPROB:th 52
theorem
for b1 being non empty natural set holds
ex b2 being FinSequence of REAL st
len b2 = b1 &
(for b3 being Element of NAT
st b3 in dom b2
holds 0 <= b2 . b3) &
Sum b2 = 1;
:: MATRPROB:attrnot 1 => MATRPROB:attr 1
definition
let a1 be FinSequence of REAL;
attr a1 is ProbFinS means
(for b1 being Element of NAT
st b1 in dom a1
holds 0 <= a1 . b1) &
Sum a1 = 1;
end;
:: MATRPROB:dfs 5
definiens
let a1 be FinSequence of REAL;
To prove
a1 is ProbFinS
it is sufficient to prove
thus (for b1 being Element of NAT
st b1 in dom a1
holds 0 <= a1 . b1) &
Sum a1 = 1;
:: MATRPROB:def 5
theorem
for b1 being FinSequence of REAL holds
b1 is ProbFinS
iff
(for b2 being Element of NAT
st b2 in dom b1
holds 0 <= b1 . b2) &
Sum b1 = 1;
:: MATRPROB:exreg 1
registration
cluster Relation-like Function-like non empty finite FinSequence-like complex-valued ext-real-valued real-valued ProbFinS FinSequence of REAL;
end;
:: MATRPROB:th 53
theorem
for b1 being non empty ProbFinS FinSequence of REAL
for b2 being Element of NAT
st b2 in dom b1
holds b1 . b2 <= 1;
:: MATRPROB:th 54
theorem
for b1 being non empty set
for b2 being non empty-yielding tabular FinSequence of b1 * holds
1 <= len b2 & 1 <= width b2;
:: MATRPROB:attrnot 2 => MATRPROB:attr 2
definition
let a1 be tabular FinSequence of REAL *;
attr a1 is m-nonnegative means
for b1, b2 being Element of NAT
st [b1,b2] in Indices a1
holds 0 <= a1 *(b1,b2);
end;
:: MATRPROB:dfs 6
definiens
let a1 be tabular FinSequence of REAL *;
To prove
a1 is m-nonnegative
it is sufficient to prove
thus for b1, b2 being Element of NAT
st [b1,b2] in Indices a1
holds 0 <= a1 *(b1,b2);
:: MATRPROB:def 6
theorem
for b1 being tabular FinSequence of REAL * holds
b1 is m-nonnegative
iff
for b2, b3 being Element of NAT
st [b2,b3] in Indices b1
holds 0 <= b1 *(b2,b3);
:: MATRPROB:attrnot 3 => MATRPROB:attr 3
definition
let a1 be tabular FinSequence of REAL *;
attr a1 is with_sum=1 means
SumAll a1 = 1;
end;
:: MATRPROB:dfs 7
definiens
let a1 be tabular FinSequence of REAL *;
To prove
a1 is with_sum=1
it is sufficient to prove
thus SumAll a1 = 1;
:: MATRPROB:def 7
theorem
for b1 being tabular FinSequence of REAL * holds
b1 is with_sum=1
iff
SumAll b1 = 1;
:: MATRPROB:attrnot 4 => MATRPROB:attr 4
definition
let a1 be tabular FinSequence of REAL *;
attr a1 is Joint_Probability means
a1 is m-nonnegative & a1 is with_sum=1;
end;
:: MATRPROB:dfs 8
definiens
let a1 be tabular FinSequence of REAL *;
To prove
a1 is Joint_Probability
it is sufficient to prove
thus a1 is m-nonnegative & a1 is with_sum=1;
:: MATRPROB:def 8
theorem
for b1 being tabular FinSequence of REAL * holds
b1 is Joint_Probability
iff
b1 is m-nonnegative & b1 is with_sum=1;
:: MATRPROB:condreg 1
registration
cluster tabular Joint_Probability -> m-nonnegative with_sum=1 (FinSequence of REAL *);
end;
:: MATRPROB:condreg 2
registration
cluster tabular m-nonnegative with_sum=1 -> Joint_Probability (FinSequence of REAL *);
end;
:: MATRPROB:th 55
theorem
for b1, b2 being non empty natural set holds
ex b3 being Matrix of b1,b2,REAL st
b3 is m-nonnegative & SumAll b3 = 1;
:: MATRPROB:exreg 2
registration
cluster Relation-like non empty-yielding Function-like finite FinSequence-like tabular FinSequence-yielding Joint_Probability FinSequence of REAL *;
end;
:: MATRPROB:funcreg 2
registration
let a1, a2 be non empty natural set;
let a3 be non empty set;
let a4 be Matrix of a1,a2,a3;
cluster a4 @ -> ;
end;
:: MATRPROB:th 56
theorem
for b1 being non empty-yielding tabular Joint_Probability FinSequence of REAL * holds
b1 @ is non empty-yielding tabular Joint_Probability FinSequence of REAL *;
:: MATRPROB:th 57
theorem
for b1 being non empty-yielding tabular Joint_Probability FinSequence of REAL *
for b2, b3 being Element of NAT
st [b2,b3] in Indices b1
holds b1 *(b2,b3) <= 1;
:: MATRPROB:attrnot 5 => MATRPROB:attr 5
definition
let a1 be tabular FinSequence of REAL *;
attr a1 is with_line_sum=1 means
for b1 being Element of NAT
st b1 in dom a1
holds Sum (a1 . b1) = 1;
end;
:: MATRPROB:dfs 9
definiens
let a1 be tabular FinSequence of REAL *;
To prove
a1 is with_line_sum=1
it is sufficient to prove
thus for b1 being Element of NAT
st b1 in dom a1
holds Sum (a1 . b1) = 1;
:: MATRPROB:def 9
theorem
for b1 being tabular FinSequence of REAL * holds
b1 is with_line_sum=1
iff
for b2 being Element of NAT
st b2 in dom b1
holds Sum (b1 . b2) = 1;
:: MATRPROB:th 58
theorem
for b1, b2 being non empty natural set holds
ex b3 being Matrix of b1,b2,REAL st
b3 is m-nonnegative & b3 is with_line_sum=1;
:: MATRPROB:attrnot 6 => MATRPROB:attr 6
definition
let a1 be tabular FinSequence of REAL *;
attr a1 is Conditional_Probability means
a1 is m-nonnegative & a1 is with_line_sum=1;
end;
:: MATRPROB:dfs 10
definiens
let a1 be tabular FinSequence of REAL *;
To prove
a1 is Conditional_Probability
it is sufficient to prove
thus a1 is m-nonnegative & a1 is with_line_sum=1;
:: MATRPROB:def 10
theorem
for b1 being tabular FinSequence of REAL * holds
b1 is Conditional_Probability
iff
b1 is m-nonnegative & b1 is with_line_sum=1;
:: MATRPROB:condreg 3
registration
cluster tabular Conditional_Probability -> m-nonnegative with_line_sum=1 (FinSequence of REAL *);
end;
:: MATRPROB:condreg 4
registration
cluster tabular m-nonnegative with_line_sum=1 -> Conditional_Probability (FinSequence of REAL *);
end;
:: MATRPROB:exreg 3
registration
cluster Relation-like non empty-yielding Function-like finite FinSequence-like tabular FinSequence-yielding Conditional_Probability FinSequence of REAL *;
end;
:: MATRPROB:th 59
theorem
for b1 being non empty-yielding tabular Conditional_Probability FinSequence of REAL *
for b2, b3 being Element of NAT
st [b2,b3] in Indices b1
holds b1 *(b2,b3) <= 1;
:: MATRPROB:th 60
theorem
for b1 being non empty-yielding tabular FinSequence of REAL * holds
b1 is non empty-yielding tabular Conditional_Probability FinSequence of REAL *
iff
for b2 being Element of NAT
st b2 in dom b1
holds Line(b1,b2) is non empty ProbFinS FinSequence of REAL;
:: MATRPROB:th 61
theorem
for b1 being non empty-yielding tabular with_line_sum=1 FinSequence of REAL * holds
SumAll b1 = len b1;
:: MATRPROB:funcnot 9 => MATRPROB:func 3
notation
let a1 be tabular FinSequence of REAL *;
synonym Row_Marginal a1 for Sum a1;
end;
:: MATRPROB:funcnot 10 => MATRPROB:func 4
notation
let a1 be tabular FinSequence of REAL *;
synonym Column_Marginal a1 for ColSum a1;
end;
:: MATRPROB:funcreg 3
registration
let a1 be non empty-yielding tabular Joint_Probability FinSequence of REAL *;
cluster Sum a1 -> non empty ProbFinS;
end;
:: MATRPROB:funcreg 4
registration
let a1 be non empty-yielding tabular Joint_Probability FinSequence of REAL *;
cluster ColSum a1 -> non empty ProbFinS;
end;
:: MATRPROB:funcreg 5
registration
let a1 be non empty-yielding tabular FinSequence of REAL *;
cluster a1 @ -> non empty-yielding tabular;
end;
:: MATRPROB:funcreg 6
registration
let a1 be non empty-yielding tabular Joint_Probability FinSequence of REAL *;
cluster a1 @ -> tabular Joint_Probability;
end;
:: MATRPROB:th 62
theorem
for b1 being non empty ProbFinS FinSequence of REAL
for b2 being non empty-yielding tabular Conditional_Probability FinSequence of REAL *
st len b1 = len b2
holds b1 * b2 is non empty ProbFinS FinSequence of REAL &
len (b1 * b2) = width b2;
:: MATRPROB:th 63
theorem
for b1, b2 being non empty-yielding tabular Conditional_Probability FinSequence of REAL *
st width b1 = len b2
holds b1 * b2 is non empty-yielding tabular Conditional_Probability FinSequence of REAL * &
len (b1 * b2) = len b1 &
width (b1 * b2) = width b2;