:: The Definition of Finite Sequences and Matrices of Probability, and
:: Addition of Matrices of Real Elements
:: by Bo Zhang and Yatsuka Nakamura
::
:: Received August 18, 2006
:: Copyright (c) 2006-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, XBOOLE_0, SUBSET_1, NAT_1, REAL_1, FINSEQ_1, RELAT_1,
FUNCT_1, TARSKI, XREAL_0, FINSEQ_2, CARD_1, XXREAL_0, SEQ_1, ARYTM_3,
CARD_3, MATRIX_1, TREES_1, ZFMISC_1, INCSP_1, MATRIXC1, QC_LANG1,
ORDINAL4, PRE_POLY, BINOP_2, FINSEQ_3, RVSUM_1, STRUCT_0, VECTSP_1,
SUPINF_2, FVSUM_1, MATRIXR1, MATRPROB, FUNCT_7, MATRIX_0, ASYMPT_1,
XCMPLX_0, VALUED_0;
notations TARSKI, SUBSET_1, XBOOLE_0, XXREAL_0, XREAL_0, XCMPLX_0, REAL_1,
ORDINAL1, NUMBERS, RELAT_1, FUNCT_1, SEQ_1, ZFMISC_1, FUNCOP_1, BINOP_1,
BINOP_2, FUNCT_2, NAT_1, VALUED_0, RVSUM_1, FINSEQ_1, FINSEQ_2, FINSEQOP,
NEWTON, STRUCT_0, MATRLIN, MATRIXR1, MATRIX_3, MATRIX_0, GROUP_1,
FVSUM_1, RLVECT_1, VECTSP_1;
constructors REAL_1, BINOP_2, NEWTON, FVSUM_1, MATRIX_3, MATRLIN, MATRIXR1,
BINOP_1, RVSUM_1, RELSET_1, FINSEQ_2, FINSEQ_4, SEQ_1;
registrations XBOOLE_0, ORDINAL1, RELSET_1, NUMBERS, XXREAL_0, NAT_1,
MEMBERED, FINSEQ_1, FINSEQ_2, STRUCT_0, VECTSP_1, MATRIX_0, MATRLIN,
VALUED_0, CARD_1, RVSUM_1, XREAL_0;
requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
begin
reserve D for non empty set,
i,j,k for Nat,
n,m for Nat,
r for Real,
e for real-valued FinSequence;
definition
let d be set, g be FinSequence of d*, n be Nat;
redefine func g.n -> FinSequence of d;
end;
definition
let x be Real;
redefine func <*x*> -> FinSequence of REAL;
end;
theorem :: MATRPROB:1
for a being Element of D, m being non zero Nat, g being
FinSequence of D holds (len g = m & for i be Nat st i in dom g holds g.i = a)
iff g = m |-> a;
theorem :: MATRPROB:2
for a,b being Element of D holds ex g be FinSequence of D st len
g = n & for i be Nat st i in Seg n holds (i in Seg k implies g.i = a) & (not i
in Seg k implies g.i = b);
theorem :: MATRPROB:3
(for i be Nat st i in dom e holds 0 <= e.i) implies for f being
Real_Sequence st (for n be Nat st 0 <> n & n < len e holds f.(n+1) = f.n+e.(n+1
)) holds for n,m be Nat st n in dom e & m in dom e & n <= m holds f.n <= f.m;
theorem :: MATRPROB:4
len e >= 1 & (for i be Nat st i in dom e holds 0 <= e.i) implies
for f being Real_Sequence st f.1 = e.1 & (for n be Nat st 0 <> n & n < len e
holds f.(n+1) = f.n+e.(n+1)) holds for n be Nat st n in dom e holds e.n <= f.n;
theorem :: MATRPROB:5
(for i be Nat st i in dom e holds 0 <= e.i) implies for k be Nat
st k in dom e holds e.k <= Sum e;
theorem :: MATRPROB:6
for r1,r2 being Real, k being Nat, seq1 being Real_Sequence holds
ex seq being Real_Sequence st seq.0=r1 & for n holds (n<>0 & n <= k implies seq
.n=seq1.n) & (n > k implies seq.n=r2);
theorem :: MATRPROB:7
for F being FinSequence of REAL
ex f being Real_Sequence st f.0 = 0 &
(for i be Nat st i < len F holds f.(i+1) = f.i+(F.(i+1))) & Sum F = f.len F;
theorem :: MATRPROB:8
for D being set, e1 being FinSequence of D holds n |-> e1 is
FinSequence of D*;
theorem :: MATRPROB:9
for D being set, e1,e2 being FinSequence of D holds ex e being
FinSequence of D* st len e = n & for i be Nat st i in Seg n holds (i in Seg k
implies e.i = e1) & (not i in Seg k implies e.i = e2);
theorem :: MATRPROB:10
for D being set, s being FinSequence holds (s is Matrix of D iff
ex n st for i st i in dom s holds ex p being FinSequence of D st s.i = p & len
p = n);
theorem :: MATRPROB:11
for D being set, e being FinSequence of D* holds (ex n st for i
st i in dom e holds len(e.i) = n) iff e is Matrix of D;
theorem :: MATRPROB:12
for M being tabular FinSequence holds [i,j] in Indices M iff i
in Seg len M & j in Seg width M;
theorem :: MATRPROB:13
for D being non empty set, M being Matrix of D holds [i,j] in
Indices M iff i in dom M & j in dom (M.i);
theorem :: MATRPROB:14
for D being non empty set, M being Matrix of D st [i,j] in
Indices M holds M*(i,j)=(M.i).j;
theorem :: MATRPROB:15
for D being non empty set, M being Matrix of D holds [i,j] in
Indices M iff i in dom Col(M,j) & j in dom Line(M,i);
theorem :: MATRPROB:16
for D1,D2 being non empty set, M1 being (Matrix of D1),M2 being
Matrix of D2 st M1 = M2 holds for i st i in dom M1 holds Line(M1,i) = Line(M2,i
);
theorem :: MATRPROB:17
for D1,D2 being non empty set,M1 being (Matrix of D1),M2 being
Matrix of D2 st M1 = M2 holds for j st j in Seg width M1 holds Col(M1,j) = Col(
M2,j);
theorem :: MATRPROB:18
for e1 being FinSequence of D st len e1 = m holds n |-> e1 is Matrix of n,m,D
;
theorem :: MATRPROB:19
for e1,e2 being FinSequence of D st len e1 = m & len e2 = m
holds ex M being Matrix of n,m,D st for i be Nat st i in Seg n holds (i in Seg
k implies M.i = e1) & (not i in Seg k implies M.i = e2);
definition
let e be FinSequence of REAL*;
func Sum e -> FinSequence of REAL means
:: MATRPROB:def 1
len it = len e & for k st k in dom it holds it.k = Sum(e.k);
end;
notation
let m be Matrix of REAL;
synonym LineSum m for Sum m;
end;
theorem :: MATRPROB:20
for m being Matrix of REAL holds len Sum m = len m & for i st i
in Seg len m holds (Sum m).i=Sum Line(m,i);
definition
let m be Matrix of REAL;
func ColSum m -> FinSequence of REAL means
:: MATRPROB:def 2
len it = width m & for j be Nat st j in Seg width m holds it.j=Sum Col(m,j);
end;
theorem :: MATRPROB:21
for M be Matrix of REAL st width M > 0 holds LineSum M = ColSum(M@);
theorem :: MATRPROB:22
for M be Matrix of REAL holds ColSum M = LineSum(M@);
definition
let M be Matrix of REAL;
func SumAll M -> Real equals
:: MATRPROB:def 3
Sum Sum M;
end;
theorem :: MATRPROB:23
for M be Matrix of REAL st len M = 0 holds SumAll M = 0;
theorem :: MATRPROB:24
for M be Matrix of m,0,REAL holds SumAll M = 0;
theorem :: MATRPROB:25
for M1 be Matrix of n,k,REAL for M2 being Matrix of m,k,REAL
holds Sum (M1^M2) = (Sum M1)^(Sum M2);
theorem :: MATRPROB:26
for M1,M2 be Matrix of REAL holds Sum M1 + Sum M2 = Sum (M1 ^^ M2);
theorem :: MATRPROB:27
for M1,M2 be Matrix of REAL st len M1 = len M2 holds SumAll M1 +
SumAll M2 = SumAll(M1 ^^ M2);
theorem :: MATRPROB:28
for M be Matrix of REAL holds SumAll M = SumAll(M@);
theorem :: MATRPROB:29
for M be Matrix of REAL holds SumAll M = Sum ColSum M;
theorem :: MATRPROB:30
for x,y being FinSequence of REAL st len x = len y holds
len mlt (x,y) = len x;
theorem :: MATRPROB:31
for i for R being Element of i-tuples_on REAL holds mlt(i|->1,R) = R;
theorem :: MATRPROB:32
for x being FinSequence of REAL holds mlt((len x|->1),x) = x;
theorem :: MATRPROB:33
for x,y being FinSequence of REAL st (for i st i in dom x holds
x.i >= 0) & (for i st i in dom y holds y.i >= 0) holds for k st k in dom mlt(x,
y) holds (mlt(x,y)).k >= 0;
theorem :: MATRPROB:34
for i for e1,e2 being Element of i-tuples_on REAL for f1,f2
being Element of i-tuples_on the carrier of F_Real st e1 = f1 & e2 = f2 holds
mlt(e1,e2) = mlt(f1,f2);
theorem :: MATRPROB:35
for e1,e2 being FinSequence of REAL for f1,f2 being FinSequence
of F_Real st len e1 = len e2 & e1 = f1 & e2 = f2 holds mlt(e1,e2) = mlt(f1,f2);
theorem :: MATRPROB:36
for e being FinSequence of REAL for f being FinSequence of
F_Real st e = f holds Sum e = Sum f;
notation
let e1,e2 be FinSequence of REAL;
synonym e1 "*" e2 for |( e1,e2 )|;
end;
theorem :: MATRPROB:37
for i for e1,e2 being Element of i-tuples_on REAL for f1,f2 being
Element of i-tuples_on the carrier of F_Real st e1 = f1 & e2 = f2 holds e1 "*"
e2 = f1 "*" f2;
theorem :: MATRPROB:38
for e1,e2 being FinSequence of REAL for f1,f2 being FinSequence
of F_Real st len e1 = len e2 & e1 = f1 & e2 = f2 holds e1 "*" e2 = f1 "*" f2;
theorem :: MATRPROB:39
for M,M1,M2 being Matrix of REAL st width M1 = len M2 holds M =
M1*M2 iff len M = len M1 & width M = width M2 & for i,j st [i,j] in Indices M
holds M*(i,j) = Line(M1,i) "*" Col(M2,j);
theorem :: MATRPROB:40
for M being Matrix of REAL for p being FinSequence of REAL st
len M = len p holds for i st i in Seg len (p*M) holds (p*M).i = p "*" Col(M,i);
theorem :: MATRPROB:41
for M being Matrix of REAL for p being FinSequence of REAL st
width M = len p & width M > 0 holds for i st i in Seg len (M*p) holds (M*p).i =
Line(M,i) "*" p;
theorem :: MATRPROB:42
for M,M1,M2 being Matrix of REAL st width M1 = len M2 holds M =
M1*M2 iff len M = len M1 & width M = width M2 & for i st i in Seg len M holds
Line(M,i) = Line(M1,i) * M2;
definition
let x,y be FinSequence of REAL,M be Matrix of REAL;
assume that
len x = len M and
len y = width M;
func QuadraticForm(x,M,y) -> Matrix of REAL means
:: MATRPROB:def 4
len it = len x &
width it = len y & for i,j be Nat st [i,j] in Indices M
holds it*(i,j)=(x.i)*(M*(i,j))*(y.j);
end;
theorem :: MATRPROB:43
for x,y being FinSequence of REAL,M being Matrix of REAL st len x =
len M & len y = width M & len y > 0 holds (QuadraticForm(x,M,y))@ =
QuadraticForm(y,M@,x);
theorem :: MATRPROB:44
for x,y being FinSequence of REAL,M being Matrix of REAL st len
x = len M & len y = width M & len y>0 holds |(x,M*y)| = SumAll QuadraticForm(x,
M,y);
theorem :: MATRPROB:45
for x being FinSequence of REAL holds |(x,(len x |-> 1))| = Sum x;
theorem :: MATRPROB:46
for x,y being FinSequence of REAL,M being Matrix of REAL st len
x = len M & len y = width M holds |(x*M,y)| = SumAll QuadraticForm(x,M,y);
theorem :: MATRPROB:47
for x,y being FinSequence of REAL,M being Matrix of REAL st len
x = len M & len y = width M & len y>0 holds |((x*M),y)| = |(x,(M*y))|;
theorem :: MATRPROB:48
for x,y being FinSequence of REAL,M being Matrix of REAL st len y=len
M & len x =width M & len x>0 & len y>0 holds |((M*x),y)| = |(x,(M@*y))|;
theorem :: MATRPROB:49
for x,y being FinSequence of REAL,M being Matrix of REAL st len
y=len M & len x =width M & len x>0 & len y>0 holds |(x,(y*M))| = |((x*M@),y)|
;
theorem :: MATRPROB:50
for x being FinSequence of REAL,M being Matrix of REAL st len x
= len M & x = len x |-> 1 holds for k st k in Seg len(x*M) holds (x*M).k = Sum
Col(M,k);
theorem :: MATRPROB:51
for x being FinSequence of REAL,M being Matrix of REAL st len x =
width M & width M > 0 & x = (len x |-> 1) holds for k st k in Seg len(M*x)
holds (M*x).k = Sum Line(M,k);
theorem :: MATRPROB:52
for n being non zero Nat ex P being FinSequence of REAL st len
P = n & (for i st i in dom P holds P.i >= 0) & Sum P = 1;
definition
let p be real-valued FinSequence;
attr p is ProbFinS means
:: MATRPROB:def 5
(for i st i in dom p holds p.i >= 0) & Sum p = 1;
end;
registration
cluster non empty ProbFinS for FinSequence of REAL;
end;
registration
cluster non empty ProbFinS for real-valued FinSequence;
end;
theorem :: MATRPROB:53
for p being non empty ProbFinS real-valued FinSequence
for k st k in dom p holds p.k <= 1;
theorem :: MATRPROB:54
for M being non empty-yielding Matrix of D holds 1<=len M & 1<= width M;
definition
let M be Matrix of REAL;
attr M is m-nonnegative means
:: MATRPROB:def 6
for i,j st [i,j] in Indices M holds M*( i,j) >= 0;
end;
definition
let M be Matrix of REAL;
attr M is with_sum=1 means
:: MATRPROB:def 7
SumAll M = 1;
end;
definition
let M be Matrix of REAL;
attr M is Joint_Probability means
:: MATRPROB:def 8
M is m-nonnegative with_sum=1;
end;
registration
cluster Joint_Probability -> m-nonnegative with_sum=1 for Matrix of REAL;
cluster m-nonnegative with_sum=1 -> Joint_Probability for Matrix of REAL;
end;
theorem :: MATRPROB:55
for n,m being non zero Nat holds ex M be Matrix of n,m,REAL st
M is m-nonnegative & SumAll M = 1;
registration
cluster non empty-yielding Joint_Probability for Matrix of REAL;
end;
theorem :: MATRPROB:56
for M being non empty-yielding Joint_Probability Matrix of REAL
holds M@ is non empty-yielding Joint_Probability Matrix of REAL;
theorem :: MATRPROB:57
for M being non empty-yielding Joint_Probability Matrix of REAL holds
for i,j st [i,j] in Indices M holds M*(i,j) <= 1;
definition
let M be Matrix of REAL;
attr M is with_line_sum=1 means
:: MATRPROB:def 9
for k st k in dom M holds Sum (M.k) = 1;
end;
theorem :: MATRPROB:58
for n,m being non zero Nat holds ex M being Matrix of n,m,REAL
st M is m-nonnegative with_line_sum=1;
definition
let M be Matrix of REAL;
attr M is Conditional_Probability means
:: MATRPROB:def 10
M is m-nonnegative with_line_sum=1;
end;
registration
cluster Conditional_Probability -> m-nonnegative with_line_sum=1 for
Matrix of
REAL;
cluster m-nonnegative with_line_sum=1 -> Conditional_Probability for
Matrix of
REAL;
end;
registration
cluster non empty-yielding Conditional_Probability for Matrix of REAL;
end;
theorem :: MATRPROB:59
for M being non empty-yielding Conditional_Probability Matrix of REAL
holds for i,j st [i,j] in Indices M holds M*(i,j) <= 1;
theorem :: MATRPROB:60
for M being non empty-yielding Matrix of REAL holds M is non
empty-yielding Conditional_Probability Matrix of REAL iff for i st i in dom M
holds Line(M,i) is non empty ProbFinS FinSequence of REAL;
theorem :: MATRPROB:61
for M being non empty-yielding with_line_sum=1 Matrix of REAL holds
SumAll M = len M;
notation
let M be Matrix of REAL;
synonym Row_Marginal M for LineSum M;
synonym Column_Marginal M for ColSum M;
end;
registration
let M be non empty-yielding Joint_Probability Matrix of REAL;
cluster Row_Marginal M -> non empty ProbFinS;
cluster Column_Marginal M -> non empty ProbFinS;
end;
registration
let M be non empty-yielding Matrix of REAL;
cluster M@ -> non empty-yielding;
end;
registration
let M be non empty-yielding Joint_Probability Matrix of REAL;
cluster M@ -> Joint_Probability;
end;
theorem :: MATRPROB:62
for p being non empty ProbFinS FinSequence of REAL for P being
non empty-yielding Conditional_Probability Matrix of REAL st len p = len P
holds p * P is non empty ProbFinS FinSequence of REAL & len (p * P) = width P
;
theorem :: MATRPROB:63
for P1, P2 being non empty-yielding Conditional_Probability Matrix of
REAL st width P1 = len P2 holds P1 * P2 is non empty-yielding
Conditional_Probability Matrix of REAL & len (P1 * P2) = len P1 & width (P1 *
P2) = width P2;