:: BOR_CANT semantic presentation
begin
definition
let D be set ;
let x, y be ext-real number ;
let a, b be Element of D;
:: original: IFGT
redefine func IFGT (x,y,a,b) -> Element of D;
coherence
IFGT (x,y,a,b) is Element of D by XXREAL_0:def_11;
end;
theorem Th1: :: BOR_CANT:1
for k being Element of NAT
for x being Element of REAL st k is odd & x > 0 & x <= 1 holds
(((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0
proof
let k be Element of NAT ; ::_thesis: for x being Element of REAL st k is odd & x > 0 & x <= 1 holds
(((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0
let x be Element of REAL ; ::_thesis: ( k is odd & x > 0 & x <= 1 implies (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0 )
assume that
A1: k is odd and
A2: x > 0 and
A3: x <= 1 ; ::_thesis: (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0
consider m being Element of NAT such that
A4: k = (2 * m) + 1 by A1, ABIAN:9;
set q = m + 1;
A5: k + 2 = (2 * (m + 1)) + 1 by A4;
consider m being Element of NAT such that
A6: k = (2 * m) + 1 by A1, ABIAN:9;
A7: for k being Element of NAT st k is even holds
(- x) |^ k > 0
proof
let k be Element of NAT ; ::_thesis: ( k is even implies (- x) |^ k > 0 )
assume k is even ; ::_thesis: (- x) |^ k > 0
then consider m being Element of NAT such that
A8: k = 2 * m by ABIAN:def_2;
defpred S1[ Element of NAT ] means (- x) |^ (2 * $1) > 0 ;
A9: S1[ 0 ] by NEWTON:4;
A10: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A11: S1[k] ; ::_thesis: S1[k + 1]
(- x) |^ (2 * (k + 1)) = (- x) |^ ((2 * k) + 2) ;
then A12: (- x) |^ (2 * (k + 1)) = ((- x) |^ (2 * k)) * ((- x) |^ 2) by NEWTON:8;
(- x) * (- x) > 0 by A2;
then (- x) |^ 2 > 0 by NEWTON:81;
hence S1[k + 1] by A11, A12; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A9, A10);
hence (- x) |^ k > 0 by A8; ::_thesis: verum
end;
A13: (x |^ (k + 2)) / ((- x) |^ (k + 1)) = x
proof
x |^ (k + 2) = x |^ ((k + 1) + 1) ;
then A14: x |^ (k + 2) = (x |^ (k + 1)) * x by NEWTON:6;
x |^ (k + 2) = x * ((- x) |^ (k + 1)) by A6, A14, POWER:1;
then (x |^ (k + 2)) / ((- x) |^ (k + 1)) = (x * ((- x) |^ (k + 1))) * (((- x) |^ (k + 1)) ") by XCMPLX_0:def_9;
then A15: (x |^ (k + 2)) / ((- x) |^ (k + 1)) = x * (((- x) |^ (k + 1)) * (((- x) |^ (k + 1)) ")) ;
((- x) |^ (k + 1)) * (((- x) |^ (k + 1)) ") = 1
proof
A16: 0 < (- x) |^ (k + 1) by A6, A7;
A17: 1 <= ((- x) |^ (k + 1)) / ((- x) |^ (k + 1)) by A6, A7, XREAL_1:181;
A18: ((- x) |^ (k + 1)) / ((- x) |^ (k + 1)) <= 1 by A16, XREAL_1:185;
((- x) |^ (k + 1)) / ((- x) |^ (k + 1)) = 1 by A17, A18, XXREAL_0:1;
hence ((- x) |^ (k + 1)) * (((- x) |^ (k + 1)) ") = 1 by XCMPLX_0:def_9; ::_thesis: verum
end;
hence (x |^ (k + 2)) / ((- x) |^ (k + 1)) = x by A15; ::_thesis: verum
end;
A19: 1 <= ((k + 2) !) / ((k + 1) !)
proof
(k + 2) ! = ((k + 1) + 1) * ((k + 1) !) by NEWTON:15;
then A20: ((k + 2) !) * (((k + 1) !) ") = ((k + 1) + 1) * (((k + 1) !) * (((k + 1) !) ")) ;
A21: 1 <= ((k + 1) !) / ((k + 1) !) by XREAL_1:181;
A22: ((k + 1) !) / ((k + 1) !) <= 1 by XREAL_1:183;
((k + 1) !) / ((k + 1) !) = 1 by A21, A22, XXREAL_0:1;
then ((k + 2) !) * (((k + 1) !) ") = ((k + 1) + 1) * 1 by A20, XCMPLX_0:def_9;
then ((k + 2) !) * (((k + 1) !) ") >= 1 by NAT_1:11;
hence 1 <= ((k + 2) !) / ((k + 1) !) by XCMPLX_0:def_9; ::_thesis: verum
end;
( (x |^ (k + 2)) / ((- x) |^ (k + 1)) <= ((k + 2) !) / ((k + 1) !) implies (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0 )
proof
assume A23: (x |^ (k + 2)) / ((- x) |^ (k + 1)) <= ((k + 2) !) / ((k + 1) !) ; ::_thesis: (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0
(x |^ (k + 2)) * (((- x) |^ (k + 1)) ") <= ((k + 2) !) / ((k + 1) !) by A23, XCMPLX_0:def_9;
then A24: (x |^ (k + 2)) * (((- x) |^ (k + 1)) ") <= (((k + 1) !) ") * ((k + 2) !) by XCMPLX_0:def_9;
(- x) |^ (k + 1) > 0 by A6, A7;
then A25: (x |^ (k + 2)) / ((k + 2) !) <= (((k + 1) !) ") / (((- x) |^ (k + 1)) ") by A24, XREAL_1:102;
A26: (((k + 1) !) ") * 1 = 1 / ((k + 1) !) by XCMPLX_0:def_9;
( (((k + 1) !) ") / (((- x) |^ (k + 1)) ") = (1 / ((k + 1) !)) * ((((- x) |^ (k + 1)) ") ") & 1 * (((k + 1) !) ") = 1 / ((k + 1) !) ) by A26, XCMPLX_0:def_9;
then A27: (((k + 1) !) ") / (((- x) |^ (k + 1)) ") = ((- x) |^ (k + 1)) / ((k + 1) !) by XCMPLX_0:def_9;
(x rExpSeq) . (k + 2) <= ((- x) |^ (k + 1)) / ((k + 1) !) by A25, A27, SIN_COS:def_5;
then (x rExpSeq) . (k + 2) <= ((- x) rExpSeq) . (k + 1) by SIN_COS:def_5;
then ((x rExpSeq) . (k + 2)) - (((- x) rExpSeq) . (k + 1)) <= (((- x) rExpSeq) . (k + 1)) - (((- x) rExpSeq) . (k + 1)) by XREAL_1:9;
then A28: ( ((x rExpSeq) . (k + 2)) - (((- x) rExpSeq) . (k + 1)) <= 0 & - (((x rExpSeq) . (k + 2)) - (((- x) rExpSeq) . (k + 1))) >= 0 ) ;
- ((x rExpSeq) . (k + 2)) = ((- x) rExpSeq) . (k + 2)
proof
defpred S1[ Element of NAT ] means - (x |^ ((2 * $1) + 1)) = (- x) |^ ((2 * $1) + 1);
A29: (- x) |^ ((2 * 0) + 1) = - x by NEWTON:5;
A30: S1[ 0 ] by A29, NEWTON:5;
A31: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A32: S1[k] ; ::_thesis: S1[k + 1]
- (x |^ ((2 * (k + 1)) + 1)) = - ((x |^ (((2 * k) + 1) + 1)) * x) by NEWTON:6;
then - (x |^ ((2 * (k + 1)) + 1)) = - (((x |^ ((2 * k) + 1)) * x) * x) by NEWTON:6;
then - (x |^ ((2 * (k + 1)) + 1)) = (((- x) |^ ((2 * k) + 1)) * (- x)) * (- x) by A32;
then - (x |^ ((2 * (k + 1)) + 1)) = ((- x) |^ (((2 * k) + 1) + 1)) * (- x) by NEWTON:6;
hence S1[k + 1] by NEWTON:6; ::_thesis: verum
end;
A33: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A30, A31);
consider m being Element of NAT such that
A34: k + 2 = (2 * m) + 1 by A5;
A35: - (x |^ (k + 2)) = (- x) |^ (k + 2) by A33, A34;
- ((x rExpSeq) . (k + 2)) = - ((x |^ (k + 2)) / ((k + 2) !)) by SIN_COS:def_5;
then - ((x rExpSeq) . (k + 2)) = - ((x |^ (k + 2)) * (((k + 2) !) ")) by XCMPLX_0:def_9;
then - ((x rExpSeq) . (k + 2)) = (- (x |^ (k + 2))) * (((k + 2) !) ") ;
then - ((x rExpSeq) . (k + 2)) = (- (x |^ (k + 2))) / ((k + 2) !) by XCMPLX_0:def_9;
hence - ((x rExpSeq) . (k + 2)) = ((- x) rExpSeq) . (k + 2) by A35, SIN_COS:def_5; ::_thesis: verum
end;
hence (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0 by A28; ::_thesis: verum
end;
hence (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0 by A3, A19, A13, XXREAL_0:2; ::_thesis: verum
end;
theorem Th2: :: BOR_CANT:2
for x being Element of REAL holds 1 + x <= exp_R . x
proof
let x be Element of REAL ; ::_thesis: 1 + x <= exp_R . x
percases ( x > 0 or x = 0 or x < 0 ) ;
supposeA1: x > 0 ; ::_thesis: 1 + x <= exp_R . x
set B2 = NAT --> (1 + x);
A2: for n being Element of NAT st x > 0 holds
(NAT --> (1 + x)) . n <= (Partial_Sums (x rExpSeq)) . (n + 1)
proof
let n be Element of NAT ; ::_thesis: ( x > 0 implies (NAT --> (1 + x)) . n <= (Partial_Sums (x rExpSeq)) . (n + 1) )
defpred S1[ Nat] means (NAT --> (1 + x)) . $1 <= (Partial_Sums (x rExpSeq)) . (1 + $1);
(Partial_Sums (x rExpSeq)) . 1 = ((Partial_Sums (x rExpSeq)) . 0) + ((x rExpSeq) . (0 + 1)) by SERIES_1:def_1;
then A3: (Partial_Sums (x rExpSeq)) . 1 = ((x rExpSeq) . 0) + ((x rExpSeq) . 1) by SERIES_1:def_1;
A4: (x rExpSeq) . 0 = (x |^ 0) / (0 !) by SIN_COS:def_5
.= 1 by NEWTON:4, NEWTON:12 ;
(x rExpSeq) . 1 = (x |^ 1) / (1 !) by SIN_COS:def_5;
then (x rExpSeq) . 1 = x by NEWTON:5, NEWTON:13;
then A5: S1[ 0 ] by A4, A3, FUNCOP_1:7;
A6: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A7: S1[k] ; ::_thesis: S1[k + 1]
A8: (Partial_Sums (x rExpSeq)) . (1 + (k + 1)) = ((Partial_Sums (x rExpSeq)) . (k + 1)) + ((x rExpSeq) . ((k + 1) + 1)) by SERIES_1:def_1;
A9: (x rExpSeq) . ((k + 1) + 1) > 0
proof
( x |^ ((k + 1) + 1) > 0 & ((k + 1) + 1) ! > 0 ) by A1, NEWTON:83;
then (x |^ ((k + 1) + 1)) / (((k + 1) + 1) !) > 0 ;
hence (x rExpSeq) . ((k + 1) + 1) > 0 by SIN_COS:def_5; ::_thesis: verum
end;
A10: 1 + x <= (Partial_Sums (x rExpSeq)) . (k + 1) by A7, FUNCOP_1:7;
(Partial_Sums (x rExpSeq)) . (k + 1) <= ((x rExpSeq) . ((k + 1) + 1)) + ((Partial_Sums (x rExpSeq)) . (k + 1)) by A9, XREAL_1:31;
then 1 + x <= ((Partial_Sums (x rExpSeq)) . (k + 1)) + ((x rExpSeq) . ((k + 1) + 1)) by A10, XXREAL_0:2;
hence S1[k + 1] by A8, FUNCOP_1:7; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A5, A6);
hence ( x > 0 implies (NAT --> (1 + x)) . n <= (Partial_Sums (x rExpSeq)) . (n + 1) ) ; ::_thesis: verum
end;
A11: for n being Element of NAT holds (NAT --> (1 + x)) . n <= ((Partial_Sums (x rExpSeq)) ^\ 1) . n
proof
let n be Element of NAT ; ::_thesis: (NAT --> (1 + x)) . n <= ((Partial_Sums (x rExpSeq)) ^\ 1) . n
(NAT --> (1 + x)) . n <= (Partial_Sums (x rExpSeq)) . (n + 1) by A1, A2;
hence (NAT --> (1 + x)) . n <= ((Partial_Sums (x rExpSeq)) ^\ 1) . n by NAT_1:def_3; ::_thesis: verum
end;
A12: lim (NAT --> (1 + x)) = (NAT --> (1 + x)) . 1 by SEQ_4:26
.= 1 + x by FUNCOP_1:7 ;
x rExpSeq is summable by SIN_COS:45;
then A13: Partial_Sums (x rExpSeq) is convergent by SERIES_1:def_2;
then A14: ( lim ((Partial_Sums (x rExpSeq)) ^\ 1) = lim (Partial_Sums (x rExpSeq)) & (Partial_Sums (x rExpSeq)) ^\ 1 is convergent ) by SEQ_4:20;
lim (NAT --> (1 + x)) <= lim ((Partial_Sums (x rExpSeq)) ^\ 1) by A13, A11, SEQ_2:18;
then lim (NAT --> (1 + x)) <= Sum (x rExpSeq) by A14, SERIES_1:def_3;
hence 1 + x <= exp_R . x by A12, SIN_COS:def_22; ::_thesis: verum
end;
suppose x = 0 ; ::_thesis: 1 + x <= exp_R . x
hence 1 + x <= exp_R . x by SIN_COS:51; ::_thesis: verum
end;
supposeA15: x < 0 ; ::_thesis: 1 + x <= exp_R . x
set y = - x;
1 - (- x) <= exp_R . (- (- x))
proof
percases ( - x <= 1 or - x > 1 ) ;
supposeA16: - x <= 1 ; ::_thesis: 1 - (- x) <= exp_R . (- (- x))
for x being Element of REAL st x > 0 & x <= 1 holds
1 - x <= exp_R . (- x)
proof
let x be Element of REAL ; ::_thesis: ( x > 0 & x <= 1 implies 1 - x <= exp_R . (- x) )
assume that
A17: x > 0 and
A18: x <= 1 ; ::_thesis: 1 - x <= exp_R . (- x)
set B2 = NAT --> (1 - x);
A19: for n being Element of NAT holds (NAT --> (1 - x)) . n <= (Partial_Sums ((- x) rExpSeq)) . (n + 1)
proof
let n be Element of NAT ; ::_thesis: (NAT --> (1 - x)) . n <= (Partial_Sums ((- x) rExpSeq)) . (n + 1)
defpred S1[ Element of NAT ] means (NAT --> (1 - x)) . $1 <= (Partial_Sums ((- x) rExpSeq)) . ($1 + 1);
(Partial_Sums ((- x) rExpSeq)) . (0 + 1) = ((Partial_Sums ((- x) rExpSeq)) . 0) + (((- x) rExpSeq) . 1) by SERIES_1:def_1;
then A20: (Partial_Sums ((- x) rExpSeq)) . (0 + 1) = (((- x) rExpSeq) . 0) + (((- x) rExpSeq) . 1) by SERIES_1:def_1;
((- x) rExpSeq) . 1 = ((- x) |^ 1) / (1 !) by SIN_COS:def_5;
then A21: ((- x) rExpSeq) . 1 = - x by NEWTON:5, NEWTON:13;
((- x) rExpSeq) . 0 = ((- x) |^ 0) / (0 !) by SIN_COS:def_5
.= 1 by NEWTON:4, NEWTON:12 ;
then A22: S1[ 0 ] by A21, A20, FUNCOP_1:7;
A23: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A24: S1[k] ; ::_thesis: S1[k + 1]
percases ( k is even or k is odd ) ;
suppose k is even ; ::_thesis: S1[k + 1]
then consider m being Element of NAT such that
A25: k = 2 * m by ABIAN:def_2;
A26: 1 - x <= (Partial_Sums ((- x) rExpSeq)) . (k + 1) by A24, FUNCOP_1:7;
A27: for k being Element of NAT st k is even & k > 0 holds
for y being Real holds (y rExpSeq) . k >= 0
proof
let k be Element of NAT ; ::_thesis: ( k is even & k > 0 implies for y being Real holds (y rExpSeq) . k >= 0 )
assume that
A28: k is even and
A29: k > 0 ; ::_thesis: for y being Real holds (y rExpSeq) . k >= 0
let y be Real; ::_thesis: (y rExpSeq) . k >= 0
percases ( y > 0 or y = 0 or y < 0 ) ;
suppose y > 0 ; ::_thesis: (y rExpSeq) . k >= 0
then y |^ k > 0 by NEWTON:83;
then (y |^ k) / (k !) > 0 ;
hence (y rExpSeq) . k >= 0 by SIN_COS:def_5; ::_thesis: verum
end;
suppose y = 0 ; ::_thesis: (y rExpSeq) . k >= 0
then A30: y |^ k = 0 by A29, NEWTON:84;
(y |^ k) / (k !) = 0 by A30;
hence (y rExpSeq) . k >= 0 by SIN_COS:def_5; ::_thesis: verum
end;
supposeA31: y < 0 ; ::_thesis: (y rExpSeq) . k >= 0
consider m being Element of NAT such that
A32: k = 2 * m by A28, ABIAN:def_2;
y |^ k = y |^ (m + m) by A32;
then y |^ k = (y |^ m) * (y |^ m) by NEWTON:8;
then A33: y |^ k = (y * y) |^ m by NEWTON:7;
y |^ k >= 0 by A31, A33, NEWTON:83;
then (y |^ k) / (k !) >= 0 ;
hence (y rExpSeq) . k >= 0 by SIN_COS:def_5; ::_thesis: verum
end;
end;
end;
A34: ((- x) rExpSeq) . (k + 2) >= 0 by A25, A27;
A35: (Partial_Sums ((- x) rExpSeq)) . (k + 1) <= ((Partial_Sums ((- x) rExpSeq)) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) by A34, XREAL_1:31;
1 - x <= ((Partial_Sums ((- x) rExpSeq)) . (k + 1)) + (((- x) rExpSeq) . ((k + 1) + 1)) by A26, A35, XXREAL_0:2;
then 1 - x <= (Partial_Sums ((- x) rExpSeq)) . (k + 2) by SERIES_1:def_1;
hence S1[k + 1] by FUNCOP_1:7; ::_thesis: verum
end;
suppose k is odd ; ::_thesis: S1[k + 1]
then consider m being Element of NAT such that
A36: k = (2 * m) + 1 by ABIAN:9;
A37: for k being Element of NAT
for x being Element of REAL st k is odd & x > 0 & x <= 1 holds
1 - x <= (Partial_Sums ((- x) rExpSeq)) . k
proof
let k be Element of NAT ; ::_thesis: for x being Element of REAL st k is odd & x > 0 & x <= 1 holds
1 - x <= (Partial_Sums ((- x) rExpSeq)) . k
let x be Element of REAL ; ::_thesis: ( k is odd & x > 0 & x <= 1 implies 1 - x <= (Partial_Sums ((- x) rExpSeq)) . k )
assume A38: k is odd ; ::_thesis: ( not x > 0 or not x <= 1 or 1 - x <= (Partial_Sums ((- x) rExpSeq)) . k )
assume A39: x > 0 ; ::_thesis: ( not x <= 1 or 1 - x <= (Partial_Sums ((- x) rExpSeq)) . k )
assume A40: x <= 1 ; ::_thesis: 1 - x <= (Partial_Sums ((- x) rExpSeq)) . k
defpred S2[ Element of NAT ] means 1 - x <= (Partial_Sums ((- x) rExpSeq)) . ((2 * $1) + 1);
(Partial_Sums ((- x) rExpSeq)) . ((2 * 0) + 1) = ((Partial_Sums ((- x) rExpSeq)) . 0) + (((- x) rExpSeq) . 1) by SERIES_1:def_1;
then A41: (Partial_Sums ((- x) rExpSeq)) . ((2 * 0) + 1) = (((- x) rExpSeq) . 0) + (((- x) rExpSeq) . 1) by SERIES_1:def_1;
A42: ((- x) rExpSeq) . 0 = ((- x) |^ 0) / (0 !) by SIN_COS:def_5
.= 1 by NEWTON:4, NEWTON:12 ;
((- x) rExpSeq) . 1 = ((- x) |^ 1) / (1 !) by SIN_COS:def_5;
then A43: ((- x) rExpSeq) . (1 + 0) = - x by NEWTON:5, NEWTON:13;
A44: S2[ 0 ] by A43, A41, A42;
A45: for k being Element of NAT st S2[k] holds
S2[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] )
assume A46: S2[k] ; ::_thesis: S2[k + 1]
(Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 1) <= (Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 3)
proof
(Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 3) = ((Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 2)) + (((- x) rExpSeq) . (((2 * k) + 2) + 1)) by SERIES_1:def_1;
then (Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 3) = ((Partial_Sums ((- x) rExpSeq)) . (((2 * k) + 1) + 1)) + (((- x) rExpSeq) . ((2 * k) + 3)) ;
then (Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 3) = (((Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 1)) + (((- x) rExpSeq) . ((2 * k) + 2))) + (((- x) rExpSeq) . ((2 * k) + 3)) by SERIES_1:def_1;
then A47: (Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 3) = ((Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 1)) + ((((- x) rExpSeq) . ((2 * k) + 2)) + (((- x) rExpSeq) . ((2 * k) + 3))) ;
(((- x) rExpSeq) . (((2 * k) + 1) + 1)) + (((- x) rExpSeq) . (((2 * k) + 1) + 2)) >= 0 by A39, A40, Th1;
hence (Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 1) <= (Partial_Sums ((- x) rExpSeq)) . ((2 * k) + 3) by A47, XREAL_1:31; ::_thesis: verum
end;
hence S2[k + 1] by A46, XXREAL_0:2; ::_thesis: verum
end;
A48: for k being Element of NAT holds S2[k] from NAT_1:sch_1(A44, A45);
consider m being Element of NAT such that
A49: k = (2 * m) + 1 by A38, ABIAN:9;
thus 1 - x <= (Partial_Sums ((- x) rExpSeq)) . k by A48, A49; ::_thesis: verum
end;
1 - x <= (Partial_Sums ((- x) rExpSeq)) . (k + 2) by A36, A17, A18, A37;
hence S1[k + 1] by FUNCOP_1:7; ::_thesis: verum
end;
end;
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A22, A23);
hence (NAT --> (1 - x)) . n <= (Partial_Sums ((- x) rExpSeq)) . (n + 1) ; ::_thesis: verum
end;
A50: for n being Element of NAT holds (NAT --> (1 - x)) . n <= ((Partial_Sums ((- x) rExpSeq)) ^\ 1) . n
proof
let n be Element of NAT ; ::_thesis: (NAT --> (1 - x)) . n <= ((Partial_Sums ((- x) rExpSeq)) ^\ 1) . n
(NAT --> (1 - x)) . n <= (Partial_Sums ((- x) rExpSeq)) . (n + 1) by A19;
hence (NAT --> (1 - x)) . n <= ((Partial_Sums ((- x) rExpSeq)) ^\ 1) . n by NAT_1:def_3; ::_thesis: verum
end;
A51: lim (NAT --> (1 - x)) = (NAT --> (1 - x)) . 1 by SEQ_4:26
.= 1 - x by FUNCOP_1:7 ;
(- x) rExpSeq is summable by SIN_COS:45;
then A52: Partial_Sums ((- x) rExpSeq) is convergent by SERIES_1:def_2;
then A53: ( lim ((Partial_Sums ((- x) rExpSeq)) ^\ 1) = lim (Partial_Sums ((- x) rExpSeq)) & (Partial_Sums ((- x) rExpSeq)) ^\ 1 is convergent ) by SEQ_4:20;
lim (NAT --> (1 - x)) <= lim ((Partial_Sums ((- x) rExpSeq)) ^\ 1) by A52, A50, SEQ_2:18;
then lim (NAT --> (1 - x)) <= Sum ((- x) rExpSeq) by A53, SERIES_1:def_3;
hence 1 - x <= exp_R . (- x) by A51, SIN_COS:def_22; ::_thesis: verum
end;
hence 1 - (- x) <= exp_R . (- (- x)) by A15, A16; ::_thesis: verum
end;
supposeA54: - x > 1 ; ::_thesis: 1 - (- x) <= exp_R . (- (- x))
0 < exp_R . (- (- x)) by A54, SIN_COS:53;
hence 1 - (- x) <= exp_R . (- (- x)) by A54, XREAL_1:49; ::_thesis: verum
end;
end;
end;
hence 1 + x <= exp_R . x ; ::_thesis: verum
end;
end;
end;
definition
let s be Real_Sequence;
func JSum s -> Real_Sequence means :Def1: :: BOR_CANT:def 1
for d being Nat holds it . d = Sum ((- (s . d)) rExpSeq);
existence
ex b1 being Real_Sequence st
for d being Nat holds b1 . d = Sum ((- (s . d)) rExpSeq)
proof
deffunc H1( Element of NAT ) -> Element of REAL = Sum ((- (s . $1)) rExpSeq);
consider f being Real_Sequence such that
A1: for d being Element of NAT holds f . d = H1(d) from FUNCT_2:sch_4();
take f ; ::_thesis: for d being Nat holds f . d = Sum ((- (s . d)) rExpSeq)
let d be Nat; ::_thesis: f . d = Sum ((- (s . d)) rExpSeq)
d in NAT by ORDINAL1:def_12;
hence f . d = Sum ((- (s . d)) rExpSeq) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being Real_Sequence st ( for d being Nat holds b1 . d = Sum ((- (s . d)) rExpSeq) ) & ( for d being Nat holds b2 . d = Sum ((- (s . d)) rExpSeq) ) holds
b1 = b2
proof
let f1, f2 be Real_Sequence; ::_thesis: ( ( for d being Nat holds f1 . d = Sum ((- (s . d)) rExpSeq) ) & ( for d being Nat holds f2 . d = Sum ((- (s . d)) rExpSeq) ) implies f1 = f2 )
assume that
A2: for d being Nat holds f1 . d = Sum ((- (s . d)) rExpSeq) and
A3: for d being Nat holds f2 . d = Sum ((- (s . d)) rExpSeq) ; ::_thesis: f1 = f2
let d be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: f1 . d = f2 . d
f1 . d = Sum ((- (s . d)) rExpSeq) by A2;
hence f1 . d = f2 . d by A3; ::_thesis: verum
end;
end;
:: deftheorem Def1 defines JSum BOR_CANT:def_1_:_
for s, b2 being Real_Sequence holds
( b2 = JSum s iff for d being Nat holds b2 . d = Sum ((- (s . d)) rExpSeq) );
theorem Th3: :: BOR_CANT:3
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n))
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n))
let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n))
let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma
for n being Element of NAT holds (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n))
let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n))
let n be Element of NAT ; ::_thesis: (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n))
defpred S1[ Element of NAT ] means exp_R . (- ((Partial_Sums (Prob * A)) . $1)) = (Partial_Product (JSum (Prob * A))) . $1;
A1: exp_R . (- ((Partial_Sums (Prob * A)) . 0)) = exp_R . (- ((Prob * A) . 0)) by SERIES_1:def_1;
(Partial_Product (JSum (Prob * A))) . 0 = (JSum (Prob * A)) . 0 by SERIES_3:def_1;
then (Partial_Product (JSum (Prob * A))) . 0 = Sum ((- ((Prob * A) . 0)) rExpSeq) by Def1;
then A2: S1[ 0 ] by A1, SIN_COS:def_22;
A3: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A4: S1[k] ; ::_thesis: S1[k + 1]
A5: (Partial_Product (JSum (Prob * A))) . (k + 1) = ((Partial_Product (JSum (Prob * A))) . k) * ((JSum (Prob * A)) . (k + 1)) by SERIES_3:def_1;
A6: (Partial_Product (JSum (Prob * A))) . (k + 1) = (exp_R . (- ((Partial_Sums (Prob * A)) . k))) * (Sum ((- ((Prob * A) . (k + 1))) rExpSeq)) by A4, A5, Def1;
A7: (exp_R (- ((Partial_Sums (Prob * A)) . k))) * (exp_R (- ((Prob * A) . (k + 1)))) = exp_R ((- ((Partial_Sums (Prob * A)) . k)) + (- ((Prob * A) . (k + 1)))) by SIN_COS:50;
- (((Partial_Sums (Prob * A)) . k) + ((Prob * A) . (k + 1))) = - ((Partial_Sums (Prob * A)) . (k + 1)) by SERIES_1:def_1;
hence S1[k + 1] by A7, A6, SIN_COS:def_22; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A2, A3);
hence (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n)) ; ::_thesis: verum
end;
theorem Th4: :: BOR_CANT:4
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n
let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n
let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma
for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n
let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n
let n be Element of NAT ; ::_thesis: (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n
defpred S1[ Element of NAT ] means (Partial_Product (Prob * (Complement A))) . $1 <= (Partial_Product (JSum (Prob * A))) . $1;
A1: (Partial_Product (Prob * (Complement A))) . 0 = (Prob * (Complement A)) . 0 by SERIES_3:def_1;
dom (Prob * (Complement A)) = NAT by FUNCT_2:def_1;
then A2: (Prob * (Complement A)) . 0 = Prob . ((Complement A) . 0) by FUNCT_1:12;
A3: (Partial_Product (Prob * (Complement A))) . 0 = Prob . ((A . 0) `) by A2, A1, PROB_1:def_2;
Prob . ((A . 0) `) = Prob . (([#] Sigma) \ (A . 0)) by SUBSET_1:def_4;
then A4: (Partial_Product (Prob * (Complement A))) . 0 = 1 - (Prob . (A . 0)) by A3, PROB_1:32;
(Partial_Product (JSum (Prob * A))) . 0 = (JSum (Prob * A)) . 0 by SERIES_3:def_1;
then (Partial_Product (JSum (Prob * A))) . 0 = Sum ((- ((Prob * A) . 0)) rExpSeq) by Def1;
then A5: (Partial_Product (JSum (Prob * A))) . 0 = exp_R . (- ((Prob * A) . 0)) by SIN_COS:def_22;
A6: dom (Prob * A) = NAT by FUNCT_2:def_1;
1 + (- (Prob . (A . 0))) <= exp_R . (- (Prob . (A . 0))) by Th2;
then A7: S1[ 0 ] by A4, A6, A5, FUNCT_1:12;
A8: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A9: S1[k] ; ::_thesis: S1[k + 1]
( Prob . ((Complement A) . (k + 1)) = Prob . ((A . (k + 1)) `) & (A . (k + 1)) ` = ([#] Sigma) \ (A . (k + 1)) ) by PROB_1:def_2, SUBSET_1:def_4;
then A10: Prob . ((Complement A) . (k + 1)) = 1 - (Prob . (A . (k + 1))) by PROB_1:32;
A11: 1 + (- (Prob . (A . (k + 1)))) <= exp_R . (- (Prob . (A . (k + 1)))) by Th2;
dom (Prob * (Complement A)) = NAT by FUNCT_2:def_1;
then A12: (Prob * (Complement A)) . (k + 1) <= exp_R . (- (Prob . (A . (k + 1)))) by A11, A10, FUNCT_1:12;
A13: ((Prob * (Complement A)) . (k + 1)) * ((Partial_Product (JSum (Prob * A))) . k) <= (exp_R . (- (Prob . (A . (k + 1))))) * ((Partial_Product (JSum (Prob * A))) . k)
proof
for n being Element of NAT holds (JSum (Prob * A)) . n > 0
proof
let n be Element of NAT ; ::_thesis: (JSum (Prob * A)) . n > 0
A14: exp_R . (- ((Prob * A) . n)) > 0 by SIN_COS:54;
(JSum (Prob * A)) . n = Sum ((- ((Prob * A) . n)) rExpSeq) by Def1;
hence (JSum (Prob * A)) . n > 0 by A14, SIN_COS:def_22; ::_thesis: verum
end;
then (Partial_Product (JSum (Prob * A))) . k > 0 by SERIES_3:43;
hence ((Prob * (Complement A)) . (k + 1)) * ((Partial_Product (JSum (Prob * A))) . k) <= (exp_R . (- (Prob . (A . (k + 1))))) * ((Partial_Product (JSum (Prob * A))) . k) by A12, XREAL_1:64; ::_thesis: verum
end;
A15: ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * (Complement A)) . (k + 1)) <= ((Partial_Product (JSum (Prob * A))) . k) * ((Prob * (Complement A)) . (k + 1))
proof
for n being Element of NAT holds (Prob * (Complement A)) . n >= 0
proof
let n be Element of NAT ; ::_thesis: (Prob * (Complement A)) . n >= 0
A16: Prob . ((Complement A) . n) >= 0 by PROB_1:def_8;
dom (Prob * (Complement A)) = NAT by FUNCT_2:def_1;
hence (Prob * (Complement A)) . n >= 0 by A16, FUNCT_1:12; ::_thesis: verum
end;
then (Prob * (Complement A)) . (k + 1) >= 0 ;
hence ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * (Complement A)) . (k + 1)) <= ((Partial_Product (JSum (Prob * A))) . k) * ((Prob * (Complement A)) . (k + 1)) by A9, XREAL_1:64; ::_thesis: verum
end;
((Partial_Product (Prob * (Complement A))) . k) * ((Prob * (Complement A)) . (k + 1)) <= (exp_R . (- (Prob . (A . (k + 1))))) * ((Partial_Product (JSum (Prob * A))) . k) by A15, A13, XXREAL_0:2;
then (Partial_Product (Prob * (Complement A))) . (k + 1) <= (exp_R . (- (Prob . (A . (k + 1))))) * ((Partial_Product (JSum (Prob * A))) . k) by SERIES_3:def_1;
then (Partial_Product (Prob * (Complement A))) . (k + 1) <= (Sum ((- (Prob . (A . (k + 1)))) rExpSeq)) * ((Partial_Product (JSum (Prob * A))) . k) by SIN_COS:def_22;
then (Partial_Product (Prob * (Complement A))) . (k + 1) <= (Sum ((- (Prob . (A . (k + 1)))) rExpSeq)) * (exp_R . (- ((Partial_Sums (Prob * A)) . k))) by Th3;
then (Partial_Product (Prob * (Complement A))) . (k + 1) <= (exp_R (- (Prob . (A . (k + 1))))) * (exp_R (- ((Partial_Sums (Prob * A)) . k))) by SIN_COS:def_22;
then A17: (Partial_Product (Prob * (Complement A))) . (k + 1) <= exp_R ((- (Prob . (A . (k + 1)))) + (- ((Partial_Sums (Prob * A)) . k))) by SIN_COS:50;
dom (Prob * A) = NAT by FUNCT_2:def_1;
then (Prob * A) . (k + 1) = Prob . (A . (k + 1)) by FUNCT_1:12;
then (- (Prob . (A . (k + 1)))) + (- ((Partial_Sums (Prob * A)) . k)) = - (((Prob * A) . (k + 1)) + ((Partial_Sums (Prob * A)) . k)) ;
then (Partial_Product (Prob * (Complement A))) . (k + 1) <= exp_R . (- ((Partial_Sums (Prob * A)) . (k + 1))) by A17, SERIES_1:def_1;
hence S1[k + 1] by Th3; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A7, A8);
hence (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n ; ::_thesis: verum
end;
definition
let n1, n2 be Element of NAT ;
func Special_Function (n1,n2) -> sequence of NAT means :Def2: :: BOR_CANT:def 2
for n being Element of NAT holds it . n = IFGT (n,n1,(n + n2),n);
existence
ex b1 being sequence of NAT st
for n being Element of NAT holds b1 . n = IFGT (n,n1,(n + n2),n)
proof
deffunc H1( Element of NAT ) -> Element of NAT = IFGT ($1,n1,($1 + n2),$1);
ex f being sequence of NAT st
for n being Element of NAT holds f . n = H1(n) from FUNCT_2:sch_4();
hence ex b1 being sequence of NAT st
for n being Element of NAT holds b1 . n = IFGT (n,n1,(n + n2),n) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being sequence of NAT st ( for n being Element of NAT holds b1 . n = IFGT (n,n1,(n + n2),n) ) & ( for n being Element of NAT holds b2 . n = IFGT (n,n1,(n + n2),n) ) holds
b1 = b2
proof
let s1, s2 be sequence of NAT; ::_thesis: ( ( for n being Element of NAT holds s1 . n = IFGT (n,n1,(n + n2),n) ) & ( for n being Element of NAT holds s2 . n = IFGT (n,n1,(n + n2),n) ) implies s1 = s2 )
assume that
A1: for n being Element of NAT holds s1 . n = IFGT (n,n1,(n + n2),n) and
A2: for n being Element of NAT holds s2 . n = IFGT (n,n1,(n + n2),n) ; ::_thesis: s1 = s2
let n be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: s1 . n = s2 . n
( s1 . n = IFGT (n,n1,(n + n2),n) & s2 . n = IFGT (n,n1,(n + n2),n) ) by A1, A2;
hence s1 . n = s2 . n ; ::_thesis: verum
end;
end;
:: deftheorem Def2 defines Special_Function BOR_CANT:def_2_:_
for n1, n2 being Element of NAT
for b3 being sequence of NAT holds
( b3 = Special_Function (n1,n2) iff for n being Element of NAT holds b3 . n = IFGT (n,n1,(n + n2),n) );
definition
let k be Element of NAT ;
func Special_Function2 k -> sequence of NAT means :Def3: :: BOR_CANT:def 3
for n being Element of NAT holds it . n = n + k;
existence
ex b1 being sequence of NAT st
for n being Element of NAT holds b1 . n = n + k
proof
deffunc H1( Element of NAT ) -> Element of NAT = $1 + k;
consider f being sequence of NAT such that
A1: for n being Element of NAT holds f . n = H1(n) from FUNCT_2:sch_4();
take f ; ::_thesis: for n being Element of NAT holds f . n = n + k
let n be Nat; ::_thesis: ( n is Element of REAL & n is Element of NAT implies f . n = n + k )
thus ( n is Element of REAL & n is Element of NAT implies f . n = n + k ) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being sequence of NAT st ( for n being Element of NAT holds b1 . n = n + k ) & ( for n being Element of NAT holds b2 . n = n + k ) holds
b1 = b2
proof
let s1, s2 be sequence of NAT; ::_thesis: ( ( for n being Element of NAT holds s1 . n = n + k ) & ( for n being Element of NAT holds s2 . n = n + k ) implies s1 = s2 )
assume that
A2: for n being Element of NAT holds s1 . n = n + k and
A3: for n being Element of NAT holds s2 . n = n + k ; ::_thesis: s1 = s2
let n be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: s1 . n = s2 . n
( s1 . n = n + k & s2 . n = n + k ) by A2, A3;
hence s1 . n = s2 . n ; ::_thesis: verum
end;
end;
:: deftheorem Def3 defines Special_Function2 BOR_CANT:def_3_:_
for k being Element of NAT
for b2 being sequence of NAT holds
( b2 = Special_Function2 k iff for n being Element of NAT holds b2 . n = n + k );
definition
let k be Element of NAT ;
func Special_Function3 k -> sequence of NAT means :Def4: :: BOR_CANT:def 4
for n being Element of NAT holds it . n = IFGT (n,k,0,1);
existence
ex b1 being sequence of NAT st
for n being Element of NAT holds b1 . n = IFGT (n,k,0,1)
proof
deffunc H1( Element of NAT ) -> Element of NAT = IFGT ($1,k,0,1);
ex f being sequence of NAT st
for n being Element of NAT holds f . n = H1(n) from FUNCT_2:sch_4();
hence ex b1 being sequence of NAT st
for n being Element of NAT holds b1 . n = IFGT (n,k,0,1) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being sequence of NAT st ( for n being Element of NAT holds b1 . n = IFGT (n,k,0,1) ) & ( for n being Element of NAT holds b2 . n = IFGT (n,k,0,1) ) holds
b1 = b2
proof
let s1, s2 be sequence of NAT; ::_thesis: ( ( for n being Element of NAT holds s1 . n = IFGT (n,k,0,1) ) & ( for n being Element of NAT holds s2 . n = IFGT (n,k,0,1) ) implies s1 = s2 )
assume that
A1: for n being Element of NAT holds s1 . n = IFGT (n,k,0,1) and
A2: for n being Element of NAT holds s2 . n = IFGT (n,k,0,1) ; ::_thesis: s1 = s2
let n be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: s1 . n = s2 . n
( s1 . n = IFGT (n,k,0,1) & s2 . n = IFGT (n,k,0,1) ) by A1, A2;
hence s1 . n = s2 . n ; ::_thesis: verum
end;
end;
:: deftheorem Def4 defines Special_Function3 BOR_CANT:def_4_:_
for k being Element of NAT
for b2 being sequence of NAT holds
( b2 = Special_Function3 k iff for n being Element of NAT holds b2 . n = IFGT (n,k,0,1) );
definition
let n1, n2 be Element of NAT ;
func Special_Function4 (n1,n2) -> sequence of NAT means :Def5: :: BOR_CANT:def 5
for n being Element of NAT holds it . n = IFGT (n,(n1 + 1),(n + n2),n);
existence
ex b1 being sequence of NAT st
for n being Element of NAT holds b1 . n = IFGT (n,(n1 + 1),(n + n2),n)
proof
deffunc H1( Element of NAT ) -> Element of NAT = IFGT ($1,(n1 + 1),($1 + n2),$1);
ex f being sequence of NAT st
for n being Element of NAT holds f . n = H1(n) from FUNCT_2:sch_4();
hence ex b1 being sequence of NAT st
for n being Element of NAT holds b1 . n = IFGT (n,(n1 + 1),(n + n2),n) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being sequence of NAT st ( for n being Element of NAT holds b1 . n = IFGT (n,(n1 + 1),(n + n2),n) ) & ( for n being Element of NAT holds b2 . n = IFGT (n,(n1 + 1),(n + n2),n) ) holds
b1 = b2
proof
let s1, s2 be sequence of NAT; ::_thesis: ( ( for n being Element of NAT holds s1 . n = IFGT (n,(n1 + 1),(n + n2),n) ) & ( for n being Element of NAT holds s2 . n = IFGT (n,(n1 + 1),(n + n2),n) ) implies s1 = s2 )
assume that
A1: for n being Element of NAT holds s1 . n = IFGT (n,(n1 + 1),(n + n2),n) and
A2: for n being Element of NAT holds s2 . n = IFGT (n,(n1 + 1),(n + n2),n) ; ::_thesis: s1 = s2
let n be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: s1 . n = s2 . n
( s1 . n = IFGT (n,(n1 + 1),(n + n2),n) & s2 . n = IFGT (n,(n1 + 1),(n + n2),n) ) by A1, A2;
hence s1 . n = s2 . n ; ::_thesis: verum
end;
end;
:: deftheorem Def5 defines Special_Function4 BOR_CANT:def_5_:_
for n1, n2 being Element of NAT
for b3 being sequence of NAT holds
( b3 = Special_Function4 (n1,n2) iff for n being Element of NAT holds b3 . n = IFGT (n,(n1 + 1),(n + n2),n) );
registration
let n1, n2 be Element of NAT ;
cluster Special_Function (n1,n2) -> one-to-one ;
coherence
Special_Function (n1,n2) is one-to-one
proof
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom (Special_Function (n1,n2)) or not x2 in dom (Special_Function (n1,n2)) or not (Special_Function (n1,n2)) . x1 = (Special_Function (n1,n2)) . x2 or x1 = x2 )
assume that
A1: x1 in dom (Special_Function (n1,n2)) and
A2: x2 in dom (Special_Function (n1,n2)) ; ::_thesis: ( not (Special_Function (n1,n2)) . x1 = (Special_Function (n1,n2)) . x2 or x1 = x2 )
assume A3: (Special_Function (n1,n2)) . x1 = (Special_Function (n1,n2)) . x2 ; ::_thesis: x1 = x2
reconsider x1 = x1 as Element of NAT by A1;
reconsider x2 = x2 as Element of NAT by A2;
A4: ( (Special_Function (n1,n2)) . x2 = IFGT (x2,n1,(x2 + n2),x2) & (Special_Function (n1,n2)) . x1 = IFGT (x1,n1,(x1 + n2),x1) ) by Def2;
percases ( ( x1 <= n1 & x2 <= n1 ) or ( x1 <= n1 & x2 > n1 ) or ( x2 <= n1 & x1 > n1 ) or ( x1 > n1 & x2 > n1 ) ) ;
supposeA5: ( x1 <= n1 & x2 <= n1 ) ; ::_thesis: x1 = x2
( IFGT (x2,n1,(x2 + n2),x2) = x2 & IFGT (x1,n1,(x1 + n2),x1) = x1 ) by A5, XXREAL_0:def_11;
hence x1 = x2 by A4, A3; ::_thesis: verum
end;
supposeA6: ( x1 <= n1 & x2 > n1 ) ; ::_thesis: x1 = x2
then IFGT (x2,n1,(x2 + n2),x2) = x2 + n2 by XXREAL_0:def_11;
then A7: (Special_Function (n1,n2)) . x2 = x2 + n2 by Def2;
A8: IFGT (x1,n1,(x1 + n2),x1) = x1 by A6, XXREAL_0:def_11;
A9: (Special_Function (n1,n2)) . x1 = x1 by Def2, A8;
( x1 <> x2 implies (Special_Function (n1,n2)) . x1 <> (Special_Function (n1,n2)) . x2 )
proof
assume x1 <> x2 ; ::_thesis: (Special_Function (n1,n2)) . x1 <> (Special_Function (n1,n2)) . x2
( x1 < x2 & 0 <= n2 ) by A6, XXREAL_0:2;
hence (Special_Function (n1,n2)) . x1 <> (Special_Function (n1,n2)) . x2 by A9, A7, XREAL_1:31; ::_thesis: verum
end;
hence x1 = x2 by A3; ::_thesis: verum
end;
supposeA10: ( x2 <= n1 & x1 > n1 ) ; ::_thesis: x1 = x2
A11: (Special_Function (n1,n2)) . x1 = IFGT (x1,n1,(x1 + n2),x1) by Def2;
A12: (Special_Function (n1,n2)) . x1 = x1 + n2 by A11, A10, XXREAL_0:def_11;
A13: (Special_Function (n1,n2)) . x2 = IFGT (x2,n1,(x2 + n2),x2) by Def2;
A14: (Special_Function (n1,n2)) . x2 = x2 by A13, A10, XXREAL_0:def_11;
( x2 <> x1 implies (Special_Function (n1,n2)) . x2 <> (Special_Function (n1,n2)) . x1 )
proof
assume x2 <> x1 ; ::_thesis: (Special_Function (n1,n2)) . x2 <> (Special_Function (n1,n2)) . x1
( x2 < x1 & 0 <= n2 ) by A10, XXREAL_0:2;
hence (Special_Function (n1,n2)) . x2 <> (Special_Function (n1,n2)) . x1 by A14, A12, XREAL_1:31; ::_thesis: verum
end;
hence x1 = x2 by A3; ::_thesis: verum
end;
supposeA15: ( x1 > n1 & x2 > n1 ) ; ::_thesis: x1 = x2
( IFGT (x2,n1,(x2 + n2),x2) = x2 + n2 & IFGT (x1,n1,(x1 + n2),x1) = x1 + n2 ) by A15, XXREAL_0:def_11;
then x2 + n2 = x1 + n2 by A4, A3;
hence x1 = x2 ; ::_thesis: verum
end;
end;
end;
cluster Special_Function4 (n1,n2) -> one-to-one ;
coherence
Special_Function4 (n1,n2) is one-to-one
proof
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom (Special_Function4 (n1,n2)) or not x2 in dom (Special_Function4 (n1,n2)) or not (Special_Function4 (n1,n2)) . x1 = (Special_Function4 (n1,n2)) . x2 or x1 = x2 )
assume that
A16: x1 in dom (Special_Function4 (n1,n2)) and
A17: x2 in dom (Special_Function4 (n1,n2)) ; ::_thesis: ( not (Special_Function4 (n1,n2)) . x1 = (Special_Function4 (n1,n2)) . x2 or x1 = x2 )
assume A18: (Special_Function4 (n1,n2)) . x1 = (Special_Function4 (n1,n2)) . x2 ; ::_thesis: x1 = x2
reconsider x1 = x1 as Element of NAT by A16;
reconsider x2 = x2 as Element of NAT by A17;
percases ( ( x1 <= n1 + 1 & x2 <= n1 + 1 ) or ( x1 > n1 + 1 & x2 <= n1 + 1 ) or ( x1 <= n1 + 1 & x2 > n1 + 1 ) or ( x1 > n1 + 1 & x2 > n1 + 1 ) ) ;
supposeA19: ( x1 <= n1 + 1 & x2 <= n1 + 1 ) ; ::_thesis: x1 = x2
A20: ( (Special_Function4 (n1,n2)) . x2 = IFGT (x2,(n1 + 1),(x2 + n2),x2) & (Special_Function4 (n1,n2)) . x1 = IFGT (x1,(n1 + 1),(x1 + n2),x1) ) by Def5;
( IFGT (x2,(n1 + 1),(x2 + n2),x2) = x2 & IFGT (x1,(n1 + 1),(x1 + n2),x1) = x1 ) by A19, XXREAL_0:def_11;
hence x1 = x2 by A20, A18; ::_thesis: verum
end;
supposeA21: ( x1 > n1 + 1 & x2 <= n1 + 1 ) ; ::_thesis: x1 = x2
A22: ( (Special_Function4 (n1,n2)) . x2 = IFGT (x2,(n1 + 1),(x2 + n2),x2) & (Special_Function4 (n1,n2)) . x1 = IFGT (x1,(n1 + 1),(x1 + n2),x1) ) by Def5;
A23: ( (Special_Function4 (n1,n2)) . x2 = x2 & (Special_Function4 (n1,n2)) . x1 = x1 + n2 ) by A22, A21, XXREAL_0:def_11;
( x1 <> x2 implies (Special_Function4 (n1,n2)) . x2 <> (Special_Function4 (n1,n2)) . x1 )
proof
assume x1 <> x2 ; ::_thesis: (Special_Function4 (n1,n2)) . x2 <> (Special_Function4 (n1,n2)) . x1
( (Special_Function4 (n1,n2)) . x1 > (n1 + 1) + n2 & (n1 + 1) + n2 >= n1 + 1 ) by A23, A21, XREAL_1:6, XREAL_1:31;
hence (Special_Function4 (n1,n2)) . x2 <> (Special_Function4 (n1,n2)) . x1 by A21, A23, XXREAL_0:2; ::_thesis: verum
end;
hence x1 = x2 by A18; ::_thesis: verum
end;
supposeA24: ( x1 <= n1 + 1 & x2 > n1 + 1 ) ; ::_thesis: x1 = x2
A25: ( (Special_Function4 (n1,n2)) . x2 = IFGT (x2,(n1 + 1),(x2 + n2),x2) & (Special_Function4 (n1,n2)) . x1 = IFGT (x1,(n1 + 1),(x1 + n2),x1) & IFGT (x2,(n1 + 1),(x2 + n2),x2) = x2 + n2 & IFGT (x1,(n1 + 1),(x1 + n2),x1) = x1 ) by Def5, A24, XXREAL_0:def_11;
( x1 <> x2 implies (Special_Function4 (n1,n2)) . x2 <> (Special_Function4 (n1,n2)) . x1 )
proof
assume x1 <> x2 ; ::_thesis: (Special_Function4 (n1,n2)) . x2 <> (Special_Function4 (n1,n2)) . x1
( (Special_Function4 (n1,n2)) . x2 > (n1 + 1) + n2 & (n1 + 1) + n2 >= n1 + 1 ) by A25, A24, XREAL_1:6, XREAL_1:31;
hence (Special_Function4 (n1,n2)) . x2 <> (Special_Function4 (n1,n2)) . x1 by A24, A25, XXREAL_0:2; ::_thesis: verum
end;
hence x1 = x2 by A18; ::_thesis: verum
end;
supposeA26: ( x1 > n1 + 1 & x2 > n1 + 1 ) ; ::_thesis: x1 = x2
A27: ( (Special_Function4 (n1,n2)) . x2 = IFGT (x2,(n1 + 1),(x2 + n2),x2) & (Special_Function4 (n1,n2)) . x1 = IFGT (x1,(n1 + 1),(x1 + n2),x1) ) by Def5;
( IFGT (x2,(n1 + 1),(x2 + n2),x2) = x2 + n2 & IFGT (x1,(n1 + 1),(x1 + n2),x1) = x1 + n2 ) by A26, XXREAL_0:def_11;
then x1 + n2 = x2 + n2 by A27, A18;
hence x1 = x2 ; ::_thesis: verum
end;
end;
end;
end;
registration
let n be Element of NAT ;
cluster Special_Function2 n -> one-to-one ;
coherence
Special_Function2 n is one-to-one
proof
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom (Special_Function2 n) or not x2 in dom (Special_Function2 n) or not (Special_Function2 n) . x1 = (Special_Function2 n) . x2 or x1 = x2 )
assume that
A1: x1 in dom (Special_Function2 n) and
A2: x2 in dom (Special_Function2 n) ; ::_thesis: ( not (Special_Function2 n) . x1 = (Special_Function2 n) . x2 or x1 = x2 )
assume A3: (Special_Function2 n) . x1 = (Special_Function2 n) . x2 ; ::_thesis: x1 = x2
reconsider x1 = x1 as Element of NAT by A1;
reconsider x2 = x2 as Element of NAT by A2;
(Special_Function2 n) . x2 = x2 + n by Def3;
then x1 + n = x2 + n by A3, Def3;
hence x1 = x2 ; ::_thesis: verum
end;
end;
registration
let Omega be non empty set ;
let Sigma be SigmaField of Omega;
let s be Element of NAT ;
let A be SetSequence of Sigma;
clusterA ^\ s -> Sigma -valued ;
coherence
A ^\ s is Sigma -valued ;
end;
theorem Th5: :: BOR_CANT:5
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for n, n1, n2 being Element of NAT holds
( ( for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds
(Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) & ( for A, B, C being SetSequence of Sigma
for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds
(Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) )
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for n, n1, n2 being Element of NAT holds
( ( for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds
(Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) & ( for A, B, C being SetSequence of Sigma
for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds
(Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) )
let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma
for n, n1, n2 being Element of NAT holds
( ( for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds
(Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) & ( for A, B, C being SetSequence of Sigma
for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds
(Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) )
let Prob be Probability of Sigma; ::_thesis: for n, n1, n2 being Element of NAT holds
( ( for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds
(Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) & ( for A, B, C being SetSequence of Sigma
for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds
(Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) )
let n, n1, n2 be Element of NAT ; ::_thesis: ( ( for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds
(Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) & ( for A, B, C being SetSequence of Sigma
for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds
(Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) )
A1: for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds
(Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1))
proof
let A, B be SetSequence of Sigma; ::_thesis: ( n > n1 & B = A * (Special_Function (n1,n2)) implies (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) )
assume that
A2: n > n1 and
A3: B = A * (Special_Function (n1,n2)) ; ::_thesis: (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1))
A4: for q being Element of NAT st q <= n1 holds
(Partial_Product (Prob * B)) . q = (Partial_Product (Prob * A)) . q
proof
let q be Element of NAT ; ::_thesis: ( q <= n1 implies (Partial_Product (Prob * B)) . q = (Partial_Product (Prob * A)) . q )
assume A5: q <= n1 ; ::_thesis: (Partial_Product (Prob * B)) . q = (Partial_Product (Prob * A)) . q
defpred S1[ Nat] means (Partial_Product (Prob * B)) . ($1 * ((Special_Function3 n1) . $1)) = (Partial_Product (Prob * A)) . ($1 * ((Special_Function3 n1) . $1));
A6: S1[ 0 ]
proof
A7: (Partial_Product (Prob * B)) . 0 = (Prob * B) . 0 by SERIES_3:def_1;
A8: (Partial_Product (Prob * A)) . 0 = (Prob * A) . 0 by SERIES_3:def_1;
dom (Prob * B) = NAT by FUNCT_2:def_1;
then A9: (Prob * B) . 0 = Prob . (B . 0) by FUNCT_1:12;
A10: dom (A * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1;
( (Special_Function (n1,n2)) . 0 = IFGT (0,n1,(0 + n2),0) & IFGT (0,n1,(0 + n2),0) = 0 ) by Def2, XXREAL_0:def_11;
then A11: (Prob * B) . 0 = Prob . (A . 0) by A10, A3, A9, FUNCT_1:12;
dom (Prob * A) = NAT by FUNCT_2:def_1;
hence S1[ 0 ] by A11, A7, A8, FUNCT_1:12; ::_thesis: verum
end;
A12: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A13: S1[k] ; ::_thesis: S1[k + 1]
percases ( k < n1 or not k < n1 ) ;
supposeA14: k < n1 ; ::_thesis: S1[k + 1]
then A15: ( (Special_Function3 n1) . k = IFGT (k,n1,0,1) & IFGT (k,n1,0,1) = 1 ) by Def4, XXREAL_0:def_11;
k + 1 <= n1 by A14, NAT_1:13;
then A16: ( (Special_Function3 n1) . (k + 1) = IFGT ((k + 1),n1,0,1) & IFGT ((k + 1),n1,0,1) = 1 ) by Def4, XXREAL_0:def_11;
A17: (Prob * B) . (k + 1) = (Prob * A) . (k + 1)
proof
dom (Prob * B) = NAT by FUNCT_2:def_1;
then A18: (Prob * B) . (k + 1) = Prob . (B . (k + 1)) by FUNCT_1:12;
A19: dom (A * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1;
k + 1 <= n1 by A14, NAT_1:13;
then ( (Special_Function (n1,n2)) . (k + 1) = IFGT ((k + 1),n1,((k + 1) + n2),(k + 1)) & IFGT ((k + 1),n1,((k + 1) + n2),(k + 1)) = k + 1 ) by Def2, XXREAL_0:def_11;
then A20: (Prob * B) . (k + 1) = Prob . (A . (k + 1)) by A19, A3, A18, FUNCT_1:12;
dom (Prob * A) = NAT by FUNCT_2:def_1;
hence (Prob * B) . (k + 1) = (Prob * A) . (k + 1) by A20, FUNCT_1:12; ::_thesis: verum
end;
(Partial_Product (Prob * B)) . (k + 1) = ((Partial_Product (Prob * A)) . k) * ((Prob * A) . (k + 1)) by A15, A13, A17, SERIES_3:def_1;
hence S1[k + 1] by A16, SERIES_3:def_1; ::_thesis: verum
end;
supposeA21: not k < n1 ; ::_thesis: S1[k + 1]
n1 < k + 1 by A21, XREAL_1:145;
then A22: ( (Special_Function3 n1) . (k + 1) = IFGT ((k + 1),n1,0,1) & IFGT ((k + 1),n1,0,1) = 0 ) by Def4, XXREAL_0:def_11;
A23: (Prob * B) . 0 = (Prob * A) . 0
proof
dom (Prob * B) = NAT by FUNCT_2:def_1;
then A24: (Prob * B) . 0 = Prob . (B . 0) by FUNCT_1:12;
A25: dom (A * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1;
( (Special_Function (n1,n2)) . 0 = IFGT (0,n1,(0 + n2),0) & IFGT (0,n1,(0 + n2),0) = 0 ) by Def2, XXREAL_0:def_11;
then A26: (Prob * B) . 0 = Prob . (A . 0) by A25, A3, A24, FUNCT_1:12;
dom (Prob * A) = NAT by FUNCT_2:def_1;
hence (Prob * B) . 0 = (Prob * A) . 0 by A26, FUNCT_1:12; ::_thesis: verum
end;
( (Partial_Product (Prob * B)) . ((k + 1) * ((Special_Function3 n1) . (k + 1))) = (Prob * B) . 0 & (Partial_Product (Prob * A)) . ((k + 1) * ((Special_Function3 n1) . (k + 1))) = (Prob * A) . 0 ) by A22, SERIES_3:def_1;
hence S1[k + 1] by A23; ::_thesis: verum
end;
end;
end;
A27: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A6, A12);
(Partial_Product (Prob * B)) . q = (Partial_Product (Prob * A)) . q
proof
( (Special_Function3 n1) . q = IFGT (q,n1,0,1) & IFGT (q,n1,0,1) = 1 ) by Def4, A5, XXREAL_0:def_11;
then q * ((Special_Function3 n1) . q) = q ;
hence (Partial_Product (Prob * B)) . q = (Partial_Product (Prob * A)) . q by A27; ::_thesis: verum
end;
hence (Partial_Product (Prob * B)) . q = (Partial_Product (Prob * A)) . q ; ::_thesis: verum
end;
defpred S1[ Element of NAT ] means (Partial_Product (Prob * B)) . (($1 + n1) + 1) = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . (((($1 + n1) + 1) - n1) - 1));
( (Partial_Product (Prob * B)) . ((0 + n1) + 1) = ((Partial_Product (Prob * B)) . n1) * ((Prob * B) . (n1 + 1)) & n1 <= n1 ) by SERIES_3:def_1;
then A28: (Partial_Product (Prob * B)) . ((0 + n1) + 1) = ((Partial_Product (Prob * A)) . n1) * ((Prob * B) . (n1 + 1)) by A4;
A29: dom (A * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1;
A30: n1 < n1 + 1 by NAT_1:13;
( (Special_Function (n1,n2)) . (n1 + 1) = IFGT ((n1 + 1),n1,((n1 + 1) + n2),(n1 + 1)) & IFGT ((n1 + 1),n1,((n1 + 1) + n2),(n1 + 1)) = (n1 + 1) + n2 ) by Def2, A30, XXREAL_0:def_11;
then A31: Prob . (B . (n1 + 1)) = Prob . (A . ((n1 + 1) + n2)) by A29, A3, FUNCT_1:12;
A32: A . (((n1 + n2) + 1) + 0) = (A ^\ ((n1 + n2) + 1)) . 0 by NAT_1:def_3;
dom (Prob * (A ^\ ((n1 + n2) + 1))) = NAT by FUNCT_2:def_1;
then A33: Prob . (B . (n1 + 1)) = (Prob * (A ^\ ((n1 + n2) + 1))) . 0 by A32, A31, FUNCT_1:12;
dom (Prob * B) = NAT by FUNCT_2:def_1;
then (Prob * B) . (n1 + 1) = (Prob * (A ^\ ((n1 + n2) + 1))) . 0 by A33, FUNCT_1:12;
then A34: S1[ 0 ] by A28, SERIES_3:def_1;
A35: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A36: S1[k] ; ::_thesis: S1[k + 1]
A37: (Partial_Product (Prob * B)) . (((k + 1) + n1) + 1) = (((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . k)) * ((Prob * B) . (((k + 1) + n1) + 1)) by A36, SERIES_3:def_1;
A38: (Prob * B) . (((k + 1) + n1) + 1) = (Prob * (A ^\ ((n1 + n2) + 1))) . (k + 1)
proof
set j = ((k + 1) + n1) + 1;
A39: dom (A * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1;
1 <= (k + 1) + 1 by XREAL_1:31;
then n1 < n1 + ((k + 1) + 1) by NAT_1:19;
then ( (Special_Function (n1,n2)) . (((k + 1) + n1) + 1) = IFGT ((((k + 1) + n1) + 1),n1,((((k + 1) + n1) + 1) + n2),(((k + 1) + n1) + 1)) & IFGT ((((k + 1) + n1) + 1),n1,((((k + 1) + n1) + 1) + n2),(((k + 1) + n1) + 1)) = (((k + 1) + n1) + 1) + n2 ) by Def2, XXREAL_0:def_11;
then B . (((k + 1) + n1) + 1) = A . (((n1 + n2) + 1) + (k + 1)) by A39, A3, FUNCT_1:12;
then A40: Prob . (B . (((k + 1) + n1) + 1)) = Prob . ((A ^\ ((n1 + n2) + 1)) . (k + 1)) by NAT_1:def_3;
dom (Prob * B) = NAT by FUNCT_2:def_1;
then A41: (Prob * B) . (((k + 1) + n1) + 1) = Prob . ((A ^\ ((n1 + n2) + 1)) . (k + 1)) by A40, FUNCT_1:12;
dom (Prob * (A ^\ ((n1 + n2) + 1))) = NAT by FUNCT_2:def_1;
hence (Prob * B) . (((k + 1) + n1) + 1) = (Prob * (A ^\ ((n1 + n2) + 1))) . (k + 1) by A41, FUNCT_1:12; ::_thesis: verum
end;
((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . k) * ((Prob * (A ^\ ((n1 + n2) + 1))) . (k + 1)) = (Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . (k + 1) by SERIES_3:def_1;
hence S1[k + 1] by A38, A37; ::_thesis: verum
end;
A42: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A34, A35);
(n - n1) - 1 is Element of NAT
proof
n1 + 1 <= n by A2, NAT_1:13;
then (n1 + 1) - 1 <= n - 1 by XREAL_1:9;
then ( n1 <= n - 1 & n - 1 is Element of NAT ) by A2, NAT_1:20;
then (n - 1) - n1 is Element of NAT by NAT_1:21;
hence (n - n1) - 1 is Element of NAT ; ::_thesis: verum
end;
then consider k being Element of NAT such that
A43: k = (n - n1) - 1 ;
(Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((((((n - n1) - 1) + n1) + 1) - n1) - 1)) by A42, A43;
hence (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ; ::_thesis: verum
end;
for A, B, C being SetSequence of Sigma
for n1, n2, n being Element of NAT
for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds
(Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))
proof
let A, B, C be SetSequence of Sigma; ::_thesis: for n1, n2, n being Element of NAT
for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds
(Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))
let n1, n2, n be Element of NAT ; ::_thesis: for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds
(Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))
let e be sequence of NAT; ::_thesis: ( n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) implies (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) )
assume A44: n > n1 ; ::_thesis: ( not C = A * e or not B = C * (Special_Function (n1,n2)) or (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) )
assume C = A * e ; ::_thesis: ( not B = C * (Special_Function (n1,n2)) or (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) )
assume A45: B = C * (Special_Function (n1,n2)) ; ::_thesis: (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))
reconsider B = B as SetSequence of Sigma ;
A46: (Partial_Intersection B) . n1 = (Partial_Intersection C) . n1
proof
for x being set holds
( x in (Partial_Intersection B) . n1 iff x in (Partial_Intersection C) . n1 )
proof
let x be set ; ::_thesis: ( x in (Partial_Intersection B) . n1 iff x in (Partial_Intersection C) . n1 )
hereby ::_thesis: ( x in (Partial_Intersection C) . n1 implies x in (Partial_Intersection B) . n1 )
assume A47: x in (Partial_Intersection B) . n1 ; ::_thesis: x in (Partial_Intersection C) . n1
x in (Partial_Intersection C) . n1
proof
A48: for knat being Nat st knat <= n1 holds
x in C . knat
proof
let knat be Nat; ::_thesis: ( knat <= n1 implies x in C . knat )
assume A49: knat <= n1 ; ::_thesis: x in C . knat
reconsider knat = knat as Element of NAT by ORDINAL1:def_12;
A50: x in B . knat by A49, A47, PROB_3:25;
A51: dom (C * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1;
( (Special_Function (n1,n2)) . knat = IFGT (knat,n1,(knat + n2),knat) & IFGT (knat,n1,(knat + n2),knat) = knat ) by Def2, A49, XXREAL_0:def_11;
hence x in C . knat by A51, A45, A50, FUNCT_1:12; ::_thesis: verum
end;
reconsider n1 = n1 as Nat ;
thus x in (Partial_Intersection C) . n1 by A48, PROB_3:25; ::_thesis: verum
end;
hence x in (Partial_Intersection C) . n1 ; ::_thesis: verum
end;
assume A52: x in (Partial_Intersection C) . n1 ; ::_thesis: x in (Partial_Intersection B) . n1
x in (Partial_Intersection B) . n1
proof
for knat being Nat st knat <= n1 holds
x in B . knat
proof
let knat be Nat; ::_thesis: ( knat <= n1 implies x in B . knat )
assume A53: knat <= n1 ; ::_thesis: x in B . knat
reconsider knat = knat as Element of NAT by ORDINAL1:def_12;
A54: x in C . knat by A53, A52, PROB_3:25;
A55: dom (C * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1;
( (Special_Function (n1,n2)) . knat = IFGT (knat,n1,(knat + n2),knat) & IFGT (knat,n1,(knat + n2),knat) = knat ) by Def2, A53, XXREAL_0:def_11;
hence x in B . knat by A55, A45, A54, FUNCT_1:12; ::_thesis: verum
end;
hence x in (Partial_Intersection B) . n1 by PROB_3:25; ::_thesis: verum
end;
hence x in (Partial_Intersection B) . n1 ; ::_thesis: verum
end;
hence (Partial_Intersection B) . n1 = (Partial_Intersection C) . n1 by TARSKI:1; ::_thesis: verum
end;
A56: for x being set st ( for knat being Nat st knat <= n holds
x in B . knat ) holds
( ( for knat being Nat st knat <= n1 holds
x in B . knat ) & ( for knat being Nat st n1 < knat & knat <= n holds
x in B . knat ) ) by A44, XXREAL_0:2;
A57: for x being set st x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) holds
for qnat being Nat st n1 < qnat & qnat <= n holds
x in B . qnat
proof
let x be set ; ::_thesis: ( x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) implies for qnat being Nat st n1 < qnat & qnat <= n holds
x in B . qnat )
assume A58: x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) ; ::_thesis: for qnat being Nat st n1 < qnat & qnat <= n holds
x in B . qnat
A59: ( (n - n1) - 1 >= 0 & (n - n1) - 1 is Element of NAT )
proof
n - n1 is Element of NAT by A44, NAT_1:21;
hence ( (n - n1) - 1 >= 0 & (n - n1) - 1 is Element of NAT ) by A44, NAT_1:20, XREAL_1:50; ::_thesis: verum
end;
A60: for knat being Nat st knat <= (n - n1) - 1 holds
x in C . (knat + ((n1 + n2) + 1))
proof
let knat be Nat; ::_thesis: ( knat <= (n - n1) - 1 implies x in C . (knat + ((n1 + n2) + 1)) )
assume knat <= (n - n1) - 1 ; ::_thesis: x in C . (knat + ((n1 + n2) + 1))
then x in (C ^\ ((n1 + n2) + 1)) . knat by A59, A58, PROB_3:25;
hence x in C . (knat + ((n1 + n2) + 1)) by NAT_1:def_3; ::_thesis: verum
end;
for qnat being Nat st n1 < qnat & qnat <= n holds
x in B . qnat
proof
let qnat be Nat; ::_thesis: ( n1 < qnat & qnat <= n implies x in B . qnat )
assume that
A61: n1 < qnat and
A62: qnat <= n ; ::_thesis: x in B . qnat
A63: ( n1 + 1 <= qnat & qnat <= n ) by A61, A62, NAT_1:13;
A64: qnat - (n1 + 1) is Element of NAT by A63, NAT_1:21;
consider knat being Nat such that
A65: knat = (qnat - n1) - 1 by A64;
A66: (qnat - n1) - 1 <= (n - n1) - 1
proof
qnat - (n1 + 1) <= n - (n1 + 1) by A62, XREAL_1:9;
hence (qnat - n1) - 1 <= (n - n1) - 1 ; ::_thesis: verum
end;
A67: x in C . (knat + ((n1 + n2) + 1)) by A65, A66, A60;
x in B . qnat
proof
reconsider qnat = qnat as Element of NAT by ORDINAL1:def_12;
A68: dom (C * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1;
( (Special_Function (n1,n2)) . qnat = IFGT (qnat,n1,(qnat + n2),qnat) & IFGT (qnat,n1,(qnat + n2),qnat) = qnat + n2 ) by Def2, A61, XXREAL_0:def_11;
hence x in B . qnat by A68, A45, A65, A67, FUNCT_1:12; ::_thesis: verum
end;
hence x in B . qnat ; ::_thesis: verum
end;
hence for qnat being Nat st n1 < qnat & qnat <= n holds
x in B . qnat ; ::_thesis: verum
end;
A69: for x being set st ( for qnat being Nat st n1 < qnat & qnat <= n holds
x in B . qnat ) holds
x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)
proof
let x be set ; ::_thesis: ( ( for qnat being Nat st n1 < qnat & qnat <= n holds
x in B . qnat ) implies x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) )
assume A70: for qnat being Nat st n1 < qnat & qnat <= n holds
x in B . qnat ; ::_thesis: x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)
A71: ( (n - n1) - 1 >= 0 & (n - n1) - 1 is Element of NAT )
proof
n - n1 is Element of NAT by A44, NAT_1:21;
hence ( (n - n1) - 1 >= 0 & (n - n1) - 1 is Element of NAT ) by A44, NAT_1:20, XREAL_1:50; ::_thesis: verum
end;
x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)
proof
A72: for qnat being Nat st 0 <= (qnat - n1) - 1 & (qnat - n1) - 1 <= (n - n1) - 1 holds
x in B . qnat
proof
let qnat be Nat; ::_thesis: ( 0 <= (qnat - n1) - 1 & (qnat - n1) - 1 <= (n - n1) - 1 implies x in B . qnat )
assume that
A73: 0 <= (qnat - n1) - 1 and
A74: (qnat - n1) - 1 <= (n - n1) - 1 ; ::_thesis: x in B . qnat
0 + (n1 + 1) <= (qnat - (n1 + 1)) + (n1 + 1) by A73, XREAL_1:6;
then (n1 + 1) - 1 <= qnat - 1 by XREAL_1:9;
then ( n1 <= qnat - 1 & qnat < qnat + 1 ) by NAT_1:13;
then ( n1 <= qnat - 1 & qnat - 1 < (qnat + 1) - 1 ) by XREAL_1:9;
then A75: n1 < qnat by XXREAL_0:2;
(qnat - (n1 + 1)) + (n1 + 1) <= (n - (n1 + 1)) + (n1 + 1) by A74, XREAL_1:6;
hence x in B . qnat by A75, A70; ::_thesis: verum
end;
for knat being Nat st 0 <= knat & knat <= (n - n1) - 1 holds
x in (C ^\ ((n1 + n2) + 1)) . knat
proof
let knat be Nat; ::_thesis: ( 0 <= knat & knat <= (n - n1) - 1 implies x in (C ^\ ((n1 + n2) + 1)) . knat )
assume that
0 <= knat and
A76: knat <= (n - n1) - 1 ; ::_thesis: x in (C ^\ ((n1 + n2) + 1)) . knat
set qnat = (knat + n1) + 1;
A77: (((knat + n1) + 1) - n1) - 1 <= (n - n1) - 1 by A76;
A78: x in B . ((knat + n1) + 1) by A77, A72;
A79: dom (C * (Special_Function (n1,n2))) = NAT by FUNCT_2:def_1;
A80: n1 < (knat + n1) + 1
proof
n1 + 1 <= (n1 + 1) + knat by XREAL_1:31;
hence n1 < (knat + n1) + 1 by NAT_1:13; ::_thesis: verum
end;
( (Special_Function (n1,n2)) . ((knat + n1) + 1) = IFGT (((knat + n1) + 1),n1,(((knat + n1) + 1) + n2),((knat + n1) + 1)) & IFGT (((knat + n1) + 1),n1,(((knat + n1) + 1) + n2),((knat + n1) + 1)) = ((knat + n1) + 1) + n2 ) by Def2, A80, XXREAL_0:def_11;
then B . ((knat + n1) + 1) = C . (((n1 + n2) + 1) + knat) by A45, A79, FUNCT_1:12;
hence x in (C ^\ ((n1 + n2) + 1)) . knat by A78, NAT_1:def_3; ::_thesis: verum
end;
then for knat being Nat st knat <= (n - n1) - 1 holds
x in (C ^\ ((n1 + n2) + 1)) . knat ;
hence x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) by A71, PROB_3:25; ::_thesis: verum
end;
hence x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) ; ::_thesis: verum
end;
A81: for x being set holds
( x in (Partial_Intersection B) . n iff ( ( for knat being Nat st knat <= n1 holds
x in B . knat ) & ( for knat being Nat st n1 < knat & knat <= n holds
x in B . knat ) ) )
proof
let x be set ; ::_thesis: ( x in (Partial_Intersection B) . n iff ( ( for knat being Nat st knat <= n1 holds
x in B . knat ) & ( for knat being Nat st n1 < knat & knat <= n holds
x in B . knat ) ) )
( x in (Partial_Intersection B) . n iff for knat being Nat st knat <= n holds
x in B . knat ) by PROB_3:25;
hence ( x in (Partial_Intersection B) . n iff ( ( for knat being Nat st knat <= n1 holds
x in B . knat ) & ( for knat being Nat st n1 < knat & knat <= n holds
x in B . knat ) ) ) by A56; ::_thesis: verum
end;
A82: for x being set holds
( x in (Partial_Intersection B) . n iff ( x in (Partial_Intersection B) . n1 & ( for knat being Nat st n1 < knat & knat <= n holds
x in B . knat ) ) )
proof
let x be set ; ::_thesis: ( x in (Partial_Intersection B) . n iff ( x in (Partial_Intersection B) . n1 & ( for knat being Nat st n1 < knat & knat <= n holds
x in B . knat ) ) )
( x in (Partial_Intersection B) . n1 iff for knat being Nat st knat <= n1 holds
x in B . knat ) by PROB_3:25;
hence ( x in (Partial_Intersection B) . n iff ( x in (Partial_Intersection B) . n1 & ( for knat being Nat st n1 < knat & knat <= n holds
x in B . knat ) ) ) by A81; ::_thesis: verum
end;
for x being set holds
( x in (Partial_Intersection B) . n iff ( x in (Partial_Intersection B) . n1 & x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) ) )
proof
let x be set ; ::_thesis: ( x in (Partial_Intersection B) . n iff ( x in (Partial_Intersection B) . n1 & x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) ) )
( x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) iff for knat being Nat st n1 < knat & knat <= n holds
x in B . knat ) by A57, A69;
hence ( x in (Partial_Intersection B) . n iff ( x in (Partial_Intersection B) . n1 & x in (Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1) ) ) by A82; ::_thesis: verum
end;
hence (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) by A46, XBOOLE_0:def_4; ::_thesis: verum
end;
hence ( ( for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds
(Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) & ( for A, B, C being SetSequence of Sigma
for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds
(Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) ) by A1; ::_thesis: verum
end;
definition
let Omega be non empty set ;
let Sigma be SigmaField of Omega;
let Prob be Probability of Sigma;
let A be SetSequence of Sigma;
predA is_all_independent_wrt Prob means :Def6: :: BOR_CANT:def 6
for B being SetSequence of Sigma st ex e being sequence of NAT st
( e is one-to-one & ( for n being Element of NAT holds A . (e . n) = B . n ) ) holds
for n being Element of NAT holds (Partial_Product (Prob * B)) . n = Prob . ((Partial_Intersection B) . n);
end;
:: deftheorem Def6 defines is_all_independent_wrt BOR_CANT:def_6_:_
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma holds
( A is_all_independent_wrt Prob iff for B being SetSequence of Sigma st ex e being sequence of NAT st
( e is one-to-one & ( for n being Element of NAT holds A . (e . n) = B . n ) ) holds
for n being Element of NAT holds (Partial_Product (Prob * B)) . n = Prob . ((Partial_Intersection B) . n) );
theorem Th6: :: BOR_CANT:6
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n, n1, n2 being Element of NAT st n > n1 & A is_all_independent_wrt Prob holds
Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1))
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n, n1, n2 being Element of NAT st n > n1 & A is_all_independent_wrt Prob holds
Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1))
let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n, n1, n2 being Element of NAT st n > n1 & A is_all_independent_wrt Prob holds
Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1))
let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma
for n, n1, n2 being Element of NAT st n > n1 & A is_all_independent_wrt Prob holds
Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1))
let A be SetSequence of Sigma; ::_thesis: for n, n1, n2 being Element of NAT st n > n1 & A is_all_independent_wrt Prob holds
Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1))
let n, n1, n2 be Element of NAT ; ::_thesis: ( n > n1 & A is_all_independent_wrt Prob implies Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) )
assume that
A1: n > n1 and
A2: A is_all_independent_wrt Prob ; ::_thesis: Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1))
A3: for A, B being SetSequence of Sigma
for k, n being Element of NAT st B = A * (Special_Function2 k) holds
(Partial_Product (Prob * (A ^\ k))) . n = (Partial_Product (Prob * B)) . n
proof
let A, B be SetSequence of Sigma; ::_thesis: for k, n being Element of NAT st B = A * (Special_Function2 k) holds
(Partial_Product (Prob * (A ^\ k))) . n = (Partial_Product (Prob * B)) . n
let k, n be Element of NAT ; ::_thesis: ( B = A * (Special_Function2 k) implies (Partial_Product (Prob * (A ^\ k))) . n = (Partial_Product (Prob * B)) . n )
assume A4: B = A * (Special_Function2 k) ; ::_thesis: (Partial_Product (Prob * (A ^\ k))) . n = (Partial_Product (Prob * B)) . n
defpred S1[ Element of NAT ] means (Partial_Product (Prob * (A ^\ k))) . $1 = (Partial_Product (Prob * B)) . $1;
dom (Prob * (A ^\ k)) = NAT by FUNCT_2:def_1;
then A5: (Prob * (A ^\ k)) . 0 = Prob . ((A ^\ k) . 0) by FUNCT_1:12;
(Prob * (A ^\ k)) . 0 = Prob . (A . (0 + k)) by A5, NAT_1:def_3;
then A6: (Partial_Product (Prob * (A ^\ k))) . 0 = Prob . (A . k) by SERIES_3:def_1;
A7: (Partial_Product (Prob * B)) . 0 = (Prob * B) . 0 by SERIES_3:def_1;
A8: (Special_Function2 k) . 0 = 0 + k by Def3;
dom (A * (Special_Function2 k)) = NAT by FUNCT_2:def_1;
then A9: Prob . (B . 0) = Prob . (A . k) by A8, A4, FUNCT_1:12;
dom (Prob * B) = NAT by FUNCT_2:def_1;
then A10: S1[ 0 ] by A9, A7, A6, FUNCT_1:12;
A11: for q being Element of NAT st S1[q] holds
S1[q + 1]
proof
let q be Element of NAT ; ::_thesis: ( S1[q] implies S1[q + 1] )
assume A12: S1[q] ; ::_thesis: S1[q + 1]
A13: (Partial_Product (Prob * (A ^\ k))) . (q + 1) = ((Partial_Product (Prob * B)) . q) * ((Prob * (A ^\ k)) . (q + 1)) by A12, SERIES_3:def_1;
(Prob * (A ^\ k)) . (q + 1) = (Prob * B) . (q + 1)
proof
dom (Prob * (A ^\ k)) = NAT by FUNCT_2:def_1;
then A14: (Prob * (A ^\ k)) . (q + 1) = Prob . ((A ^\ k) . (q + 1)) by FUNCT_1:12;
dom (Prob * B) = NAT by FUNCT_2:def_1;
then A15: (Prob * B) . (q + 1) = Prob . (B . (q + 1)) by FUNCT_1:12;
dom (A * (Special_Function2 k)) = NAT by FUNCT_2:def_1;
then A16: B . (q + 1) = A . ((Special_Function2 k) . (q + 1)) by A4, FUNCT_1:12;
( (Special_Function2 k) . (q + 1) = (q + 1) + k & (q + 1) + k = (q + 1) + k ) by Def3;
hence (Prob * (A ^\ k)) . (q + 1) = (Prob * B) . (q + 1) by A16, A15, A14, NAT_1:def_3; ::_thesis: verum
end;
hence S1[q + 1] by A13, SERIES_3:def_1; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A10, A11);
hence (Partial_Product (Prob * (A ^\ k))) . n = (Partial_Product (Prob * B)) . n ; ::_thesis: verum
end;
A17: for m, m1, m2 being Element of NAT
for e being sequence of NAT
for C, B being SetSequence of Sigma st m1 < m & e is one-to-one & C = A * e & B = C * (Special_Function (m1,m2)) holds
Prob . ((Partial_Intersection B) . m) = ((Partial_Product (Prob * C)) . m1) * ((Partial_Product (Prob * (C ^\ ((m1 + m2) + 1)))) . ((m - m1) - 1))
proof
let m, m1, m2 be Element of NAT ; ::_thesis: for e being sequence of NAT
for C, B being SetSequence of Sigma st m1 < m & e is one-to-one & C = A * e & B = C * (Special_Function (m1,m2)) holds
Prob . ((Partial_Intersection B) . m) = ((Partial_Product (Prob * C)) . m1) * ((Partial_Product (Prob * (C ^\ ((m1 + m2) + 1)))) . ((m - m1) - 1))
let e be sequence of NAT; ::_thesis: for C, B being SetSequence of Sigma st m1 < m & e is one-to-one & C = A * e & B = C * (Special_Function (m1,m2)) holds
Prob . ((Partial_Intersection B) . m) = ((Partial_Product (Prob * C)) . m1) * ((Partial_Product (Prob * (C ^\ ((m1 + m2) + 1)))) . ((m - m1) - 1))
let C, B be SetSequence of Sigma; ::_thesis: ( m1 < m & e is one-to-one & C = A * e & B = C * (Special_Function (m1,m2)) implies Prob . ((Partial_Intersection B) . m) = ((Partial_Product (Prob * C)) . m1) * ((Partial_Product (Prob * (C ^\ ((m1 + m2) + 1)))) . ((m - m1) - 1)) )
assume that
A18: m1 < m and
A19: e is one-to-one and
A20: C = A * e and
A21: B = C * (Special_Function (m1,m2)) ; ::_thesis: Prob . ((Partial_Intersection B) . m) = ((Partial_Product (Prob * C)) . m1) * ((Partial_Product (Prob * (C ^\ ((m1 + m2) + 1)))) . ((m - m1) - 1))
( B is SetSequence of Sigma & e * (Special_Function (m1,m2)) is sequence of NAT & e * (Special_Function (m1,m2)) is one-to-one & ( for n being Element of NAT holds A . ((e * (Special_Function (m1,m2))) . n) = B . n ) )
proof
for n being Element of NAT holds A . ((e * (Special_Function (m1,m2))) . n) = B . n
proof
let n be Element of NAT ; ::_thesis: A . ((e * (Special_Function (m1,m2))) . n) = B . n
A22: dom ((A * e) * (Special_Function (m1,m2))) = NAT by FUNCT_2:def_1;
A23: B . n = (A * e) . ((Special_Function (m1,m2)) . n) by A21, A20, A22, FUNCT_1:12;
dom (A * e) = NAT by FUNCT_2:def_1;
then A24: B . n = A . (e . ((Special_Function (m1,m2)) . n)) by A23, FUNCT_1:12;
dom (e * (Special_Function (m1,m2))) = NAT by FUNCT_2:def_1;
hence A . ((e * (Special_Function (m1,m2))) . n) = B . n by A24, FUNCT_1:12; ::_thesis: verum
end;
hence ( B is SetSequence of Sigma & e * (Special_Function (m1,m2)) is sequence of NAT & e * (Special_Function (m1,m2)) is one-to-one & ( for n being Element of NAT holds A . ((e * (Special_Function (m1,m2))) . n) = B . n ) ) by A19, FUNCT_1:24; ::_thesis: verum
end;
then Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * B)) . m by A2, Def6;
hence Prob . ((Partial_Intersection B) . m) = ((Partial_Product (Prob * C)) . m1) * ((Partial_Product (Prob * (C ^\ ((m1 + m2) + 1)))) . ((m - m1) - 1)) by A18, A21, Th5; ::_thesis: verum
end;
A25: for m, m1 being Element of NAT
for e being sequence of NAT
for C, B being SetSequence of Sigma st C = A * e & e is one-to-one & B = C * (Special_Function2 m1) holds
Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * (C ^\ m1))) . m
proof
let m, m1 be Element of NAT ; ::_thesis: for e being sequence of NAT
for C, B being SetSequence of Sigma st C = A * e & e is one-to-one & B = C * (Special_Function2 m1) holds
Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * (C ^\ m1))) . m
let e be sequence of NAT; ::_thesis: for C, B being SetSequence of Sigma st C = A * e & e is one-to-one & B = C * (Special_Function2 m1) holds
Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * (C ^\ m1))) . m
let C, B be SetSequence of Sigma; ::_thesis: ( C = A * e & e is one-to-one & B = C * (Special_Function2 m1) implies Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * (C ^\ m1))) . m )
assume that
A26: C = A * e and
A27: e is one-to-one and
A28: B = C * (Special_Function2 m1) ; ::_thesis: Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * (C ^\ m1))) . m
A29: ( B is SetSequence of Sigma & Special_Function2 m1 is sequence of NAT & dom (e * (Special_Function2 m1)) <> {} & e * (Special_Function2 m1) is one-to-one & ( for n being Element of NAT holds A . ((e * (Special_Function2 m1)) . n) = B . n ) )
proof
A30: for n being Element of NAT holds A . ((e * (Special_Function2 m1)) . n) = B . n
proof
let n be Element of NAT ; ::_thesis: A . ((e * (Special_Function2 m1)) . n) = B . n
dom (A * (e * (Special_Function2 m1))) = NAT by FUNCT_2:def_1;
then A31: (A * (e * (Special_Function2 m1))) . n = A . ((e * (Special_Function2 m1)) . n) by FUNCT_1:12;
dom (A * e) = NAT by FUNCT_2:def_1;
then A32: (A * e) . ((Special_Function2 m1) . n) = A . (e . ((Special_Function2 m1) . n)) by FUNCT_1:12;
dom (e * (Special_Function2 m1)) = NAT by FUNCT_2:def_1;
then A33: (A * e) . ((Special_Function2 m1) . n) = (A * (e * (Special_Function2 m1))) . n by A32, A31, FUNCT_1:12;
dom ((A * e) * (Special_Function2 m1)) = NAT by FUNCT_2:def_1;
then A34: B . n = (A * (e * (Special_Function2 m1))) . n by A33, A28, A26, FUNCT_1:12;
dom (A * (e * (Special_Function2 m1))) = NAT by FUNCT_2:def_1;
hence A . ((e * (Special_Function2 m1)) . n) = B . n by A34, FUNCT_1:12; ::_thesis: verum
end;
thus ( B is SetSequence of Sigma & Special_Function2 m1 is sequence of NAT & dom (e * (Special_Function2 m1)) <> {} & e * (Special_Function2 m1) is one-to-one & ( for n being Element of NAT holds A . ((e * (Special_Function2 m1)) . n) = B . n ) ) by A27, A30, FUNCT_1:24; ::_thesis: verum
end;
Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * B)) . m by A2, A29, Def6;
hence Prob . ((Partial_Intersection B) . m) = (Partial_Product (Prob * (C ^\ m1))) . m by A28, A3; ::_thesis: verum
end;
A35: for q being Element of NAT holds Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . q)
proof
let q be Element of NAT ; ::_thesis: Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . q)
defpred S1[ Element of NAT ] means for e being sequence of NAT
for q, n2 being Element of NAT
for C being SetSequence of Sigma st e is one-to-one & C = A * e holds
Prob . (((Partial_Intersection (Complement C)) . $1) /\ ((Partial_Intersection (C ^\ (($1 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . $1) * ((Partial_Product (Prob * (C ^\ (($1 + n2) + 1)))) . q);
A36: S1[ 0 ]
proof
let e be sequence of NAT; ::_thesis: for q, n2 being Element of NAT
for C being SetSequence of Sigma st e is one-to-one & C = A * e holds
Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q)
let q, n2 be Element of NAT ; ::_thesis: for C being SetSequence of Sigma st e is one-to-one & C = A * e holds
Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q)
let C be SetSequence of Sigma; ::_thesis: ( e is one-to-one & C = A * e implies Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q) )
assume A37: e is one-to-one ; ::_thesis: ( not C = A * e or Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q) )
assume A38: C = A * e ; ::_thesis: Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q)
Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = Prob . (((Complement C) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) by PROB_3:21;
then Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = Prob . (((C . 0) `) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) by PROB_1:def_2;
then Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = Prob . ((Omega \ (C . 0)) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) by SUBSET_1:def_4;
then A39: Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = Prob . ((Omega /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) \ ((C . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q))) by XBOOLE_1:111;
A40: Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = Prob . (((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q) \ ((C . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q))) by A39, XBOOLE_1:28;
A41: Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = (Prob . ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) - (Prob . ((C . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q))) by A40, PROB_1:33, XBOOLE_1:17;
consider m1 being Element of NAT such that
A42: m1 = 0 ;
consider m being Element of NAT such that
A43: m = (m1 + 1) + q ;
consider m2 being Element of NAT such that
A44: m2 = n2 ;
consider B being SetSequence of Omega such that
A45: B = C * (Special_Function (m1,m2)) ;
reconsider B = B as SetSequence of Sigma by A45;
A46: ( m1 < m & C = A * e & B = C * (Special_Function (m1,m2)) )
proof
( m1 < m1 + 1 & m1 + 1 <= (m1 + 1) + q ) by NAT_1:13, XREAL_1:31;
hence ( m1 < m & C = A * e & B = C * (Special_Function (m1,m2)) ) by A43, A38, A45, XXREAL_0:2; ::_thesis: verum
end;
then Prob . ((Partial_Intersection B) . m) = Prob . (((Partial_Intersection C) . m1) /\ ((Partial_Intersection (C ^\ ((m1 + m2) + 1))) . ((m - m1) - 1))) by Th5;
then ((Partial_Product (Prob * C)) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q) = Prob . (((Partial_Intersection C) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) by A46, A37, A17, A42, A44, A43;
then (Prob . ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) - (((Partial_Product (Prob * C)) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q)) = (Prob . ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) - (Prob . ((C . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q))) by PROB_3:21;
then A47: Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = (Prob . ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) - (((Prob * C) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q)) by A41, SERIES_3:def_1;
(Prob * C) . 0 = 1 - ((Prob * (Complement C)) . 0)
proof
( C . 0 = ((C . 0) `) ` & ((C . 0) `) ` = Omega \ ((C . 0) `) ) by SUBSET_1:def_4;
then ( Prob . (C . 0) = Prob . (([#] Sigma) \ ((C . 0) `)) & (C . 0) ` is Event of Sigma ) by PROB_1:20;
then A48: Prob . (C . 0) = 1 - (Prob . ((C . 0) `)) by PROB_1:32;
dom (Prob * C) = NAT by FUNCT_2:def_1;
then A49: (Prob * C) . 0 = 1 - (Prob . ((C . 0) `)) by A48, FUNCT_1:12;
dom (Prob * (Complement C)) = NAT by FUNCT_2:def_1;
then (Prob * (Complement C)) . 0 = Prob . ((Complement C) . 0) by FUNCT_1:12;
hence (Prob * C) . 0 = 1 - ((Prob * (Complement C)) . 0) by A49, PROB_1:def_2; ::_thesis: verum
end;
then A50: Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = (Prob . ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) - (((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q) - (((Prob * (Complement C)) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q))) by A47;
set m1 = (0 + n2) + 1;
set m = q;
set B = C * (Special_Function2 ((0 + n2) + 1));
reconsider B = C * (Special_Function2 ((0 + n2) + 1)) as SetSequence of Sigma ;
A51: for A, B, C being SetSequence of Sigma
for k, n being Element of NAT
for e being sequence of NAT st C = A * e & B = C * (Special_Function2 k) holds
(Partial_Intersection (C ^\ k)) . n = (Partial_Intersection B) . n
proof
let A, B, C be SetSequence of Sigma; ::_thesis: for k, n being Element of NAT
for e being sequence of NAT st C = A * e & B = C * (Special_Function2 k) holds
(Partial_Intersection (C ^\ k)) . n = (Partial_Intersection B) . n
let k, n be Element of NAT ; ::_thesis: for e being sequence of NAT st C = A * e & B = C * (Special_Function2 k) holds
(Partial_Intersection (C ^\ k)) . n = (Partial_Intersection B) . n
let e be sequence of NAT; ::_thesis: ( C = A * e & B = C * (Special_Function2 k) implies (Partial_Intersection (C ^\ k)) . n = (Partial_Intersection B) . n )
assume C = A * e ; ::_thesis: ( not B = C * (Special_Function2 k) or (Partial_Intersection (C ^\ k)) . n = (Partial_Intersection B) . n )
assume A52: B = C * (Special_Function2 k) ; ::_thesis: (Partial_Intersection (C ^\ k)) . n = (Partial_Intersection B) . n
A53: for x being set holds
( ( for knat being Nat st knat <= n holds
x in (C ^\ k) . knat ) iff for knat being Nat st knat <= n holds
x in B . knat )
proof
let x be set ; ::_thesis: ( ( for knat being Nat st knat <= n holds
x in (C ^\ k) . knat ) iff for knat being Nat st knat <= n holds
x in B . knat )
hereby ::_thesis: ( ( for knat being Nat st knat <= n holds
x in B . knat ) implies for knat being Nat st knat <= n holds
x in (C ^\ k) . knat )
assume A54: for knat being Nat st knat <= n holds
x in (C ^\ k) . knat ; ::_thesis: for knat being Nat st knat <= n holds
x in B . knat
thus for knat being Nat st knat <= n holds
x in B . knat ::_thesis: verum
proof
let knat be Nat; ::_thesis: ( knat <= n implies x in B . knat )
assume A55: knat <= n ; ::_thesis: x in B . knat
reconsider knat = knat as Element of NAT by ORDINAL1:def_12;
dom (C * (Special_Function2 k)) = NAT by FUNCT_2:def_1;
then A56: (C * (Special_Function2 k)) . knat = C . ((Special_Function2 k) . knat) by FUNCT_1:12;
( (Special_Function2 k) . knat = knat + k & knat + k = knat + k ) by Def3;
then ( x in B . knat iff x in (C ^\ k) . knat ) by A52, A56, NAT_1:def_3;
hence x in B . knat by A55, A54; ::_thesis: verum
end;
end;
assume A57: for knat being Nat st knat <= n holds
x in B . knat ; ::_thesis: for knat being Nat st knat <= n holds
x in (C ^\ k) . knat
thus for knat being Nat st knat <= n holds
x in (C ^\ k) . knat ::_thesis: verum
proof
let knat be Nat; ::_thesis: ( knat <= n implies x in (C ^\ k) . knat )
assume A58: knat <= n ; ::_thesis: x in (C ^\ k) . knat
reconsider knat = knat as Element of NAT by ORDINAL1:def_12;
dom (C * (Special_Function2 k)) = NAT by FUNCT_2:def_1;
then A59: (C * (Special_Function2 k)) . knat = C . ((Special_Function2 k) . knat) by FUNCT_1:12;
( (Special_Function2 k) . knat = knat + k & knat + k = knat + k ) by Def3;
then ( x in B . knat iff x in (C ^\ k) . knat ) by A52, A59, NAT_1:def_3;
hence x in (C ^\ k) . knat by A57, A58; ::_thesis: verum
end;
end;
for x being set holds
( x in (Partial_Intersection (C ^\ k)) . n iff x in (Partial_Intersection B) . n )
proof
let x be set ; ::_thesis: ( x in (Partial_Intersection (C ^\ k)) . n iff x in (Partial_Intersection B) . n )
( ( x in (Partial_Intersection (C ^\ k)) . n implies for knat being Nat st knat <= n holds
x in (C ^\ k) . knat ) & ( ( for knat being Nat st knat <= n holds
x in (C ^\ k) . knat ) implies x in (Partial_Intersection (C ^\ k)) . n ) & ( x in (Partial_Intersection B) . n implies for knat being Nat st knat <= n holds
x in B . knat ) & ( ( for knat being Nat st knat <= n holds
x in B . knat ) implies x in (Partial_Intersection B) . n ) ) by PROB_3:25;
hence ( x in (Partial_Intersection (C ^\ k)) . n iff x in (Partial_Intersection B) . n ) by A53; ::_thesis: verum
end;
hence (Partial_Intersection (C ^\ k)) . n = (Partial_Intersection B) . n by TARSKI:1; ::_thesis: verum
end;
A60: Prob . ((Partial_Intersection B) . q) = (Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q by A38, A37, A25;
Prob . ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q) = (Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q by A38, A51, A60;
hence Prob . (((Partial_Intersection (Complement C)) . 0) /\ ((Partial_Intersection (C ^\ ((0 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . 0) * ((Partial_Product (Prob * (C ^\ ((0 + n2) + 1)))) . q) by A50, SERIES_3:def_1; ::_thesis: verum
end;
A61: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A62: S1[k] ; ::_thesis: S1[k + 1]
let e be sequence of NAT; ::_thesis: for q, n2 being Element of NAT
for C being SetSequence of Sigma st e is one-to-one & C = A * e holds
Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)
let q, n2 be Element of NAT ; ::_thesis: for C being SetSequence of Sigma st e is one-to-one & C = A * e holds
Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)
let C be SetSequence of Sigma; ::_thesis: ( e is one-to-one & C = A * e implies Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) )
assume A63: e is one-to-one ; ::_thesis: ( not C = A * e or Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) )
assume A64: C = A * e ; ::_thesis: Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)
Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = Prob . ((((Partial_Intersection (Complement C)) . k) /\ ((Complement C) . (k + 1))) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) by PROB_3:21;
then Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = Prob . ((((C . (k + 1)) `) /\ ((Partial_Intersection (Complement C)) . k)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) by PROB_1:def_2;
then Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = Prob . (((C . (k + 1)) `) /\ (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q))) by XBOOLE_1:16;
then Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = Prob . ((Omega \ (C . (k + 1))) /\ (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q))) by SUBSET_1:def_4;
then A65: Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = Prob . ((Omega /\ (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q))) \ ((C . (k + 1)) /\ (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)))) by XBOOLE_1:50;
A66: Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = Prob . ((((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) \ ((C . (k + 1)) /\ (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)))) by A65, XBOOLE_1:28;
A67: Prob . (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . k) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)
proof
Prob . (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . k) * ((Partial_Product (Prob * (C ^\ ((k + (1 + n2)) + 1)))) . q) by A63, A64, A62;
hence Prob . (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . k) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) ; ::_thesis: verum
end;
A68: Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = (((Partial_Product (Prob * (Complement C))) . k) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)) - ((((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (Complement C))) . k)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q))
proof
(((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (Complement C))) . k)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) = Prob . ((C . (k + 1)) /\ (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)))
proof
consider F being SetSequence of Omega such that
A69: F = C * (Special_Function4 (k,n2)) ;
F is SetSequence of Sigma
proof
for n being Element of NAT holds F . n is Event of Sigma
proof
let n be Element of NAT ; ::_thesis: F . n is Event of Sigma
A70: dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1;
F . n = C . ((Special_Function4 (k,n2)) . n) by A69, A70, FUNCT_1:12;
hence F . n is Event of Sigma ; ::_thesis: verum
end;
hence F is SetSequence of Sigma by PROB_1:25; ::_thesis: verum
end;
then reconsider F = F as SetSequence of Sigma ;
( e * (Special_Function4 (k,n2)) is one-to-one & dom (e * (Special_Function4 (k,n2))) <> {} ) by A63, FUNCT_1:24;
then consider f being sequence of NAT such that
A71: ( f = e * (Special_Function4 (k,n2)) & f is one-to-one & dom f <> {} ) ;
A72: for q being set st q in NAT holds
F . q = (A * f) . q
proof
let q be set ; ::_thesis: ( q in NAT implies F . q = (A * f) . q )
assume q in NAT ; ::_thesis: F . q = (A * f) . q
then reconsider q = q as Element of NAT ;
dom (A * e) = NAT by FUNCT_2:def_1;
then A73: (A * e) . ((Special_Function4 (k,n2)) . q) = A . (e . ((Special_Function4 (k,n2)) . q)) by FUNCT_1:12;
dom ((A * e) * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1;
then A74: ((A * e) * (Special_Function4 (k,n2))) . q = A . (e . ((Special_Function4 (k,n2)) . q)) by A73, FUNCT_1:12;
dom (e * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1;
then A75: ((A * e) * (Special_Function4 (k,n2))) . q = A . ((e * (Special_Function4 (k,n2))) . q) by A74, FUNCT_1:12;
dom (A * f) = NAT by FUNCT_2:def_1;
hence F . q = (A * f) . q by A71, A75, A64, A69, FUNCT_1:12; ::_thesis: verum
end;
A76: Prob . (((Partial_Intersection (Complement F)) . k) /\ ((Partial_Intersection (F ^\ ((k + 0) + 1))) . (q + 1))) = ((Partial_Product (Prob * (Complement F))) . k) * ((Partial_Product (Prob * (F ^\ ((k + 0) + 1)))) . (q + 1)) by A71, A72, A62, FUNCT_2:12;
A77: (Partial_Intersection (Complement C)) . k = (Partial_Intersection (Complement F)) . k
proof
A78: for x being set
for knat being Nat st knat <= k holds
( x in (Complement C) . knat iff x in (Complement F) . knat )
proof
let x be set ; ::_thesis: for knat being Nat st knat <= k holds
( x in (Complement C) . knat iff x in (Complement F) . knat )
let knat be Nat; ::_thesis: ( knat <= k implies ( x in (Complement C) . knat iff x in (Complement F) . knat ) )
assume knat <= k ; ::_thesis: ( x in (Complement C) . knat iff x in (Complement F) . knat )
then A79: knat <= k + 1 by NAT_1:13;
reconsider knat = knat as Element of NAT by ORDINAL1:def_12;
dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1;
then A80: (C * (Special_Function4 (k,n2))) . knat = C . ((Special_Function4 (k,n2)) . knat) by FUNCT_1:12;
( (Special_Function4 (k,n2)) . knat = IFGT (knat,(k + 1),(knat + n2),knat) & IFGT (knat,(k + 1),(knat + n2),knat) = knat ) by Def5, A79, XXREAL_0:def_11;
then (Complement F) . knat = (C . knat) ` by A69, A80, PROB_1:def_2;
hence ( x in (Complement C) . knat iff x in (Complement F) . knat ) by PROB_1:def_2; ::_thesis: verum
end;
A81: for x being set holds
( ( for knat being Nat st knat <= k holds
x in (Complement C) . knat ) iff for knat being Nat st knat <= k holds
x in (Complement F) . knat )
proof
let x be set ; ::_thesis: ( ( for knat being Nat st knat <= k holds
x in (Complement C) . knat ) iff for knat being Nat st knat <= k holds
x in (Complement F) . knat )
hereby ::_thesis: ( ( for knat being Nat st knat <= k holds
x in (Complement F) . knat ) implies for knat being Nat st knat <= k holds
x in (Complement C) . knat )
assume A82: for knat being Nat st knat <= k holds
x in (Complement C) . knat ; ::_thesis: for knat being Nat st knat <= k holds
x in (Complement F) . knat
thus for knat being Nat st knat <= k holds
x in (Complement F) . knat ::_thesis: verum
proof
let knat be Nat; ::_thesis: ( knat <= k implies x in (Complement F) . knat )
assume A83: knat <= k ; ::_thesis: x in (Complement F) . knat
then ( x in (Complement C) . knat iff x in (Complement F) . knat ) by A78;
hence x in (Complement F) . knat by A83, A82; ::_thesis: verum
end;
end;
assume A84: for knat being Nat st knat <= k holds
x in (Complement F) . knat ; ::_thesis: for knat being Nat st knat <= k holds
x in (Complement C) . knat
thus for knat being Nat st knat <= k holds
x in (Complement C) . knat ::_thesis: verum
proof
let knat be Nat; ::_thesis: ( knat <= k implies x in (Complement C) . knat )
assume A85: knat <= k ; ::_thesis: x in (Complement C) . knat
then ( x in (Complement C) . knat iff x in (Complement F) . knat ) by A78;
hence x in (Complement C) . knat by A85, A84; ::_thesis: verum
end;
end;
for x being set holds
( x in (Partial_Intersection (Complement C)) . k iff x in (Partial_Intersection (Complement F)) . k )
proof
let x be set ; ::_thesis: ( x in (Partial_Intersection (Complement C)) . k iff x in (Partial_Intersection (Complement F)) . k )
( x in (Partial_Intersection (Complement C)) . k iff for knat being Nat st knat <= k holds
x in (Complement C) . knat ) by PROB_3:25;
then ( x in (Partial_Intersection (Complement C)) . k iff for knat being Nat st knat <= k holds
x in (Complement F) . knat ) by A81;
hence ( x in (Partial_Intersection (Complement C)) . k iff x in (Partial_Intersection (Complement F)) . k ) by PROB_3:25; ::_thesis: verum
end;
hence (Partial_Intersection (Complement C)) . k = (Partial_Intersection (Complement F)) . k by TARSKI:1; ::_thesis: verum
end;
A86: (Partial_Intersection (F ^\ (k + 1))) . (q + 1) = (C . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)
proof
A87: for x being set
for knat being Nat st knat <= q holds
( x in (F ^\ ((k + 1) + 1)) . knat iff x in (C ^\ (((k + 1) + n2) + 1)) . knat )
proof
let x be set ; ::_thesis: for knat being Nat st knat <= q holds
( x in (F ^\ ((k + 1) + 1)) . knat iff x in (C ^\ (((k + 1) + n2) + 1)) . knat )
let knat be Nat; ::_thesis: ( knat <= q implies ( x in (F ^\ ((k + 1) + 1)) . knat iff x in (C ^\ (((k + 1) + n2) + 1)) . knat ) )
assume knat <= q ; ::_thesis: ( x in (F ^\ ((k + 1) + 1)) . knat iff x in (C ^\ (((k + 1) + n2) + 1)) . knat )
reconsider knat = knat as Element of NAT by ORDINAL1:def_12;
A88: dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1;
set j = ((knat + k) + 1) + 1;
((knat + k) + 1) + 1 > k + 1
proof
( k + 1 < (k + 1) + 1 & k + 2 <= (k + 2) + knat ) by NAT_1:12, NAT_1:13;
hence ((knat + k) + 1) + 1 > k + 1 by XXREAL_0:2; ::_thesis: verum
end;
then ( (Special_Function4 (k,n2)) . (((knat + k) + 1) + 1) = IFGT ((((knat + k) + 1) + 1),(k + 1),((((knat + k) + 1) + 1) + n2),(((knat + k) + 1) + 1)) & IFGT ((((knat + k) + 1) + 1),(k + 1),((((knat + k) + 1) + 1) + n2),(((knat + k) + 1) + 1)) = (((knat + k) + 1) + 1) + n2 ) by Def5, XXREAL_0:def_11;
then F . (knat + ((k + 1) + 1)) = C . (knat + (((k + 1) + n2) + 1)) by A69, A88, FUNCT_1:12;
then (F ^\ ((k + 1) + 1)) . knat = C . (knat + (((k + 1) + n2) + 1)) by NAT_1:def_3;
hence ( x in (F ^\ ((k + 1) + 1)) . knat iff x in (C ^\ (((k + 1) + n2) + 1)) . knat ) by NAT_1:def_3; ::_thesis: verum
end;
A89: for x being set holds
( ( for knat being Nat st knat <= q holds
x in (C ^\ (((k + 1) + n2) + 1)) . knat ) iff for knat being Nat st knat <= q holds
x in (F ^\ ((k + 1) + 1)) . knat )
proof
let x be set ; ::_thesis: ( ( for knat being Nat st knat <= q holds
x in (C ^\ (((k + 1) + n2) + 1)) . knat ) iff for knat being Nat st knat <= q holds
x in (F ^\ ((k + 1) + 1)) . knat )
hereby ::_thesis: ( ( for knat being Nat st knat <= q holds
x in (F ^\ ((k + 1) + 1)) . knat ) implies for knat being Nat st knat <= q holds
x in (C ^\ (((k + 1) + n2) + 1)) . knat )
assume A90: for knat being Nat st knat <= q holds
x in (C ^\ (((k + 1) + n2) + 1)) . knat ; ::_thesis: for knat being Nat st knat <= q holds
x in (F ^\ ((k + 1) + 1)) . knat
thus for knat being Nat st knat <= q holds
x in (F ^\ ((k + 1) + 1)) . knat ::_thesis: verum
proof
let knat be Nat; ::_thesis: ( knat <= q implies x in (F ^\ ((k + 1) + 1)) . knat )
assume A91: knat <= q ; ::_thesis: x in (F ^\ ((k + 1) + 1)) . knat
then ( x in (C ^\ (((k + 1) + n2) + 1)) . knat iff x in (F ^\ ((k + 1) + 1)) . knat ) by A87;
hence x in (F ^\ ((k + 1) + 1)) . knat by A91, A90; ::_thesis: verum
end;
end;
assume A92: for knat being Nat st knat <= q holds
x in (F ^\ ((k + 1) + 1)) . knat ; ::_thesis: for knat being Nat st knat <= q holds
x in (C ^\ (((k + 1) + n2) + 1)) . knat
thus for knat being Nat st knat <= q holds
x in (C ^\ (((k + 1) + n2) + 1)) . knat ::_thesis: verum
proof
let knat be Nat; ::_thesis: ( knat <= q implies x in (C ^\ (((k + 1) + n2) + 1)) . knat )
assume A93: knat <= q ; ::_thesis: x in (C ^\ (((k + 1) + n2) + 1)) . knat
then ( x in (C ^\ (((k + 1) + n2) + 1)) . knat iff x in (F ^\ ((k + 1) + 1)) . knat ) by A87;
hence x in (C ^\ (((k + 1) + n2) + 1)) . knat by A93, A92; ::_thesis: verum
end;
end;
A94: for x being set holds
( x in (Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q iff x in (Partial_Intersection (F ^\ ((k + 1) + 1))) . q )
proof
let x be set ; ::_thesis: ( x in (Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q iff x in (Partial_Intersection (F ^\ ((k + 1) + 1))) . q )
( x in (Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q iff for knat being Nat st knat <= q holds
x in (C ^\ (((k + 1) + n2) + 1)) . knat ) by PROB_3:25;
then ( x in (Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q iff for knat being Nat st knat <= q holds
x in (F ^\ ((k + 1) + 1)) . knat ) by A89;
hence ( x in (Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q iff x in (Partial_Intersection (F ^\ ((k + 1) + 1))) . q ) by PROB_3:25; ::_thesis: verum
end;
((Partial_Intersection (F ^\ ((k + 1) + 1))) . q) /\ (C . (k + 1)) = (Partial_Intersection (F ^\ (k + 1))) . (q + 1)
proof
defpred S2[ Element of NAT ] means ((Partial_Intersection (F ^\ ((k + 1) + 1))) . $1) /\ (C . (k + 1)) = (Partial_Intersection (F ^\ (k + 1))) . ($1 + 1);
A95: S2[ 0 ]
proof
((Partial_Intersection (F ^\ ((k + 1) + 1))) . 0) /\ (C . (k + 1)) = ((F ^\ ((k + 1) + 1)) . 0) /\ (C . (k + 1)) by PROB_3:21;
then ((Partial_Intersection (F ^\ ((k + 1) + 1))) . 0) /\ (C . (k + 1)) = (F . (0 + ((k + 1) + 1))) /\ (C . (k + 1)) by NAT_1:def_3;
then A96: ((Partial_Intersection (F ^\ ((k + 1) + 1))) . 0) /\ (C . (k + 1)) = ((F ^\ (k + 1)) . (0 + 1)) /\ (C . (k + 1)) by NAT_1:def_3;
A97: dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1;
( (Special_Function4 (k,n2)) . (k + 1) = IFGT ((k + 1),(k + 1),((k + 1) + n2),(k + 1)) & IFGT ((k + 1),(k + 1),((k + 1) + n2),(k + 1)) = k + 1 ) by Def5, XXREAL_0:def_11;
then ((Partial_Intersection (F ^\ ((k + 1) + 1))) . 0) /\ (C . (k + 1)) = ((F ^\ (k + 1)) . (0 + 1)) /\ (F . (0 + (k + 1))) by A69, A97, A96, FUNCT_1:12;
then ((Partial_Intersection (F ^\ ((k + 1) + 1))) . 0) /\ (C . (k + 1)) = ((F ^\ (k + 1)) . (0 + 1)) /\ ((F ^\ (k + 1)) . 0) by NAT_1:def_3;
then ((Partial_Intersection (F ^\ ((k + 1) + 1))) . 0) /\ (C . (k + 1)) = ((Partial_Intersection (F ^\ (k + 1))) . 0) /\ ((F ^\ (k + 1)) . (0 + 1)) by PROB_3:21;
hence S2[ 0 ] by PROB_3:21; ::_thesis: verum
end;
A98: for q being Element of NAT st S2[q] holds
S2[q + 1]
proof
let q be Element of NAT ; ::_thesis: ( S2[q] implies S2[q + 1] )
assume A99: S2[q] ; ::_thesis: S2[q + 1]
((Partial_Intersection (F ^\ ((k + 1) + 1))) . (q + 1)) /\ (C . (k + 1)) = (((Partial_Intersection (F ^\ ((k + 1) + 1))) . q) /\ ((F ^\ ((k + 1) + 1)) . (q + 1))) /\ (C . (k + 1)) by PROB_3:21;
then A100: ((Partial_Intersection (F ^\ ((k + 1) + 1))) . (q + 1)) /\ (C . (k + 1)) = ((Partial_Intersection (F ^\ (k + 1))) . (q + 1)) /\ ((F ^\ ((k + 1) + 1)) . (q + 1)) by A99, XBOOLE_1:16;
(F ^\ ((k + 1) + 1)) . (q + 1) = (F ^\ (k + 1)) . ((q + 1) + 1)
proof
(F ^\ ((k + 1) + 1)) . (q + 1) = F . ((q + 1) + ((k + 1) + 1)) by NAT_1:def_3;
then (F ^\ ((k + 1) + 1)) . (q + 1) = F . (((q + 1) + 1) + (k + 1)) ;
hence (F ^\ ((k + 1) + 1)) . (q + 1) = (F ^\ (k + 1)) . ((q + 1) + 1) by NAT_1:def_3; ::_thesis: verum
end;
hence S2[q + 1] by A100, PROB_3:21; ::_thesis: verum
end;
for k being Element of NAT holds S2[k] from NAT_1:sch_1(A95, A98);
hence ((Partial_Intersection (F ^\ ((k + 1) + 1))) . q) /\ (C . (k + 1)) = (Partial_Intersection (F ^\ (k + 1))) . (q + 1) ; ::_thesis: verum
end;
hence (Partial_Intersection (F ^\ (k + 1))) . (q + 1) = (C . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q) by A94, TARSKI:1; ::_thesis: verum
end;
A101: (Partial_Product (Prob * (F ^\ (k + 1)))) . (q + 1) = ((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)
proof
defpred S2[ Element of NAT ] means (Partial_Product (Prob * (F ^\ (k + 1)))) . ($1 + 1) = ((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . $1);
A102: S2[ 0 ]
proof
A103: (F ^\ (k + 1)) . (0 + 1) = (C * (Special_Function4 (k,n2))) . ((k + 1) + 1) by A69, NAT_1:def_3;
A104: dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1;
set j = (k + 1) + 1;
(k + 1) + 1 > k + 1 by NAT_1:13;
then ( (Special_Function4 (k,n2)) . ((k + 1) + 1) = IFGT (((k + 1) + 1),(k + 1),(((k + 1) + 1) + n2),((k + 1) + 1)) & IFGT (((k + 1) + 1),(k + 1),(((k + 1) + 1) + n2),((k + 1) + 1)) = ((k + 1) + 1) + n2 ) by Def5, XXREAL_0:def_11;
then (F ^\ (k + 1)) . (0 + 1) = C . (0 + (((k + 1) + n2) + 1)) by A104, A103, FUNCT_1:12;
then A105: Prob . ((F ^\ (k + 1)) . (0 + 1)) = Prob . ((C ^\ (((k + 1) + n2) + 1)) . 0) by NAT_1:def_3;
( dom (Prob * (F ^\ (k + 1))) = NAT & dom (Prob * (C ^\ (((k + 1) + n2) + 1))) = NAT ) by FUNCT_2:def_1;
then ( (Prob * (F ^\ (k + 1))) . (0 + 1) = Prob . ((C ^\ (((k + 1) + n2) + 1)) . 0) & Prob . ((F ^\ (k + 1)) . (0 + 1)) = (Prob * (C ^\ (((k + 1) + n2) + 1))) . 0 & Prob . ((F ^\ (k + 1)) . (0 + 1)) = Prob . ((C ^\ (((k + 1) + n2) + 1)) . 0) ) by A105, FUNCT_1:12;
then A106: ((Partial_Product (Prob * (F ^\ (k + 1)))) . 0) * ((Prob * (F ^\ (k + 1))) . (0 + 1)) = ((Prob * (F ^\ (k + 1))) . 0) * ((Prob * (C ^\ (((k + 1) + n2) + 1))) . 0) by SERIES_3:def_1;
(Prob * (F ^\ (k + 1))) . 0 = (Prob * C) . (k + 1)
proof
A107: (F ^\ (k + 1)) . 0 = F . (0 + (k + 1)) by NAT_1:def_3;
A108: dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1;
A109: F . (k + 1) = C . ((Special_Function4 (k,n2)) . (k + 1)) by A69, A108, FUNCT_1:12;
A110: ( (Special_Function4 (k,n2)) . (k + 1) = IFGT ((k + 1),(k + 1),((k + 1) + n2),(k + 1)) & IFGT ((k + 1),(k + 1),((k + 1) + n2),(k + 1)) = k + 1 ) by Def5, XXREAL_0:def_11;
dom (Prob * C) = NAT by FUNCT_2:def_1;
then A111: Prob . ((F ^\ (k + 1)) . 0) = (Prob * C) . (k + 1) by A110, A109, A107, FUNCT_1:12;
dom (Prob * (F ^\ (k + 1))) = NAT by FUNCT_2:def_1;
hence (Prob * (F ^\ (k + 1))) . 0 = (Prob * C) . (k + 1) by A111, FUNCT_1:12; ::_thesis: verum
end;
then (Partial_Product (Prob * (F ^\ (k + 1)))) . (0 + 1) = ((Prob * C) . (k + 1)) * ((Prob * (C ^\ (((k + 1) + n2) + 1))) . 0) by A106, SERIES_3:def_1;
hence S2[ 0 ] by SERIES_3:def_1; ::_thesis: verum
end;
A112: for q being Element of NAT st S2[q] holds
S2[q + 1]
proof
let q be Element of NAT ; ::_thesis: ( S2[q] implies S2[q + 1] )
assume A113: S2[q] ; ::_thesis: S2[q + 1]
A114: (Prob * (F ^\ (k + 1))) . ((q + 1) + 1) = (Prob * (C ^\ (((k + 1) + n2) + 1))) . (q + 1)
proof
A115: (F ^\ (k + 1)) . ((q + 1) + 1) = (C * (Special_Function4 (k,n2))) . (((q + 1) + 1) + (k + 1)) by A69, NAT_1:def_3;
A116: dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1;
set j = ((q + 1) + 1) + (k + 1);
((q + 1) + 1) + (k + 1) > k + 1
proof
( k + 1 < (k + 1) + 1 & (k + 1) + 1 <= ((k + 1) + 1) + (q + 1) ) by NAT_1:13, XREAL_1:31;
hence ((q + 1) + 1) + (k + 1) > k + 1 by XXREAL_0:2; ::_thesis: verum
end;
then ( (Special_Function4 (k,n2)) . (((q + 1) + 1) + (k + 1)) = IFGT ((((q + 1) + 1) + (k + 1)),(k + 1),((((q + 1) + 1) + (k + 1)) + n2),(((q + 1) + 1) + (k + 1))) & IFGT ((((q + 1) + 1) + (k + 1)),(k + 1),((((q + 1) + 1) + (k + 1)) + n2),(((q + 1) + 1) + (k + 1))) = (((q + 1) + 1) + (k + 1)) + n2 ) by Def5, XXREAL_0:def_11;
then (F ^\ (k + 1)) . ((q + 1) + 1) = C . ((q + 1) + (((k + 1) + n2) + 1)) by A116, A115, FUNCT_1:12;
then A117: Prob . ((F ^\ (k + 1)) . ((q + 1) + 1)) = Prob . ((C ^\ (((k + 1) + n2) + 1)) . (q + 1)) by NAT_1:def_3;
( dom (Prob * (F ^\ (k + 1))) = NAT & dom (Prob * (C ^\ (((k + 1) + n2) + 1))) = NAT ) by FUNCT_2:def_1;
then ( (Prob * (F ^\ (k + 1))) . ((q + 1) + 1) = Prob . ((C ^\ (((k + 1) + n2) + 1)) . (q + 1)) & Prob . ((F ^\ (k + 1)) . ((q + 1) + 1)) = (Prob * (C ^\ (((k + 1) + n2) + 1))) . (q + 1) & Prob . ((F ^\ (k + 1)) . ((q + 1) + 1)) = Prob . ((C ^\ (((k + 1) + n2) + 1)) . (q + 1)) ) by A117, FUNCT_1:12;
hence (Prob * (F ^\ (k + 1))) . ((q + 1) + 1) = (Prob * (C ^\ (((k + 1) + n2) + 1))) . (q + 1) ; ::_thesis: verum
end;
(Partial_Product (Prob * (F ^\ (k + 1)))) . ((q + 1) + 1) = (((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)) * ((Prob * (C ^\ (((k + 1) + n2) + 1))) . (q + 1)) by A113, A114, SERIES_3:def_1;
then (Partial_Product (Prob * (F ^\ (k + 1)))) . ((q + 1) + 1) = ((Prob * C) . (k + 1)) * (((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) * ((Prob * (C ^\ (((k + 1) + n2) + 1))) . (q + 1))) ;
hence S2[q + 1] by SERIES_3:def_1; ::_thesis: verum
end;
for k being Element of NAT holds S2[k] from NAT_1:sch_1(A102, A112);
hence (Partial_Product (Prob * (F ^\ (k + 1)))) . (q + 1) = ((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) ; ::_thesis: verum
end;
defpred S2[ Element of NAT ] means ( ( for k being Element of NAT st k <= $1 holds
C . k = F . k ) implies (Partial_Product (Prob * (Complement F))) . $1 = (Partial_Product (Prob * (Complement C))) . $1 );
dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1;
then A118: (C * (Special_Function4 (k,n2))) . 0 = C . ((Special_Function4 (k,n2)) . 0) by FUNCT_1:12;
A119: IFGT (0,(k + 1),(0 + n2),0) = 0 by XXREAL_0:def_11;
then (F . 0) ` = (C . 0) ` by Def5, A118, A69;
then (Complement F) . 0 = (C . 0) ` by PROB_1:def_2;
then ( Prob . ((Complement F) . 0) = Prob . ((Complement C) . 0) & dom (Prob * (Complement F)) = NAT & dom (Prob * (Complement C)) = NAT ) by FUNCT_2:def_1, PROB_1:def_2;
then ( Prob . ((Complement F) . 0) = Prob . ((Complement C) . 0) & (Prob * (Complement F)) . 0 = Prob . ((Complement F) . 0) & (Prob * (Complement C)) . 0 = Prob . ((Complement C) . 0) ) by FUNCT_1:12;
then A120: ( (Partial_Product (Prob * (Complement F))) . 0 = (Prob * (Complement C)) . 0 & F . 0 = C . 0 ) by A119, Def5, A118, A69, SERIES_3:def_1;
A121: S2[ 0 ] by A120, SERIES_3:def_1;
A122: for q being Element of NAT st S2[q] holds
S2[q + 1]
proof
let q be Element of NAT ; ::_thesis: ( S2[q] implies S2[q + 1] )
assume A123: S2[q] ; ::_thesis: S2[q + 1]
A124: ( ( for k being Element of NAT st k <= q + 1 holds
C . k = F . k ) implies for k being Element of NAT st k <= q holds
C . k = F . k )
proof
assume A125: for k being Element of NAT st k <= q + 1 holds
C . k = F . k ; ::_thesis: for k being Element of NAT st k <= q holds
C . k = F . k
let k be Element of NAT ; ::_thesis: ( k <= q implies C . k = F . k )
assume k <= q ; ::_thesis: C . k = F . k
then k <= q + 1 by NAT_1:13;
hence C . k = F . k by A125; ::_thesis: verum
end;
( ( for k being Element of NAT st k <= q + 1 holds
C . k = F . k ) implies (Partial_Product (Prob * (Complement F))) . (q + 1) = (Partial_Product (Prob * (Complement C))) . (q + 1) )
proof
assume A126: for k being Element of NAT st k <= q + 1 holds
C . k = F . k ; ::_thesis: (Partial_Product (Prob * (Complement F))) . (q + 1) = (Partial_Product (Prob * (Complement C))) . (q + 1)
then ( q + 1 <= q + 1 implies (C . (q + 1)) ` = (F . (q + 1)) ` ) ;
then ( q + 1 <= q + 1 implies (Complement C) . (q + 1) = (F . (q + 1)) ` ) by PROB_1:def_2;
then A127: ((Partial_Product (Prob * (Complement F))) . q) * (Prob . ((Complement F) . (q + 1))) = ((Partial_Product (Prob * (Complement C))) . q) * (Prob . ((Complement C) . (q + 1))) by A126, A124, A123, PROB_1:def_2;
( dom (Prob * (Complement C)) = NAT & dom (Prob * (Complement F)) = NAT ) by FUNCT_2:def_1;
then ( (Prob * (Complement C)) . (q + 1) = Prob . ((Complement C) . (q + 1)) & (Prob * (Complement F)) . (q + 1) = Prob . ((Complement F) . (q + 1)) ) by FUNCT_1:12;
then (Partial_Product (Prob * (Complement F))) . (q + 1) = ((Partial_Product (Prob * (Complement C))) . q) * ((Prob * (Complement C)) . (q + 1)) by A127, SERIES_3:def_1;
hence (Partial_Product (Prob * (Complement F))) . (q + 1) = (Partial_Product (Prob * (Complement C))) . (q + 1) by SERIES_3:def_1; ::_thesis: verum
end;
hence S2[q + 1] ; ::_thesis: verum
end;
A128: for k being Element of NAT holds S2[k] from NAT_1:sch_1(A121, A122);
for q being Element of NAT st q <= k holds
C . q = F . q
proof
let q be Element of NAT ; ::_thesis: ( q <= k implies C . q = F . q )
assume q <= k ; ::_thesis: C . q = F . q
then A129: q <= k + 1 by NAT_1:13;
A130: dom (C * (Special_Function4 (k,n2))) = NAT by FUNCT_2:def_1;
( (Special_Function4 (k,n2)) . q = IFGT (q,(k + 1),(q + n2),q) & IFGT (q,(k + 1),(q + n2),q) = q ) by Def5, A129, XXREAL_0:def_11;
hence C . q = F . q by A130, A69, FUNCT_1:12; ::_thesis: verum
end;
then Prob . (((Partial_Intersection (Complement C)) . k) /\ ((C . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q))) = ((Partial_Product (Prob * (Complement C))) . k) * (((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)) by A128, A101, A86, A77, A76;
hence (((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (Complement C))) . k)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) = Prob . ((C . (k + 1)) /\ (((Partial_Intersection (Complement C)) . k) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q))) by XBOOLE_1:16; ::_thesis: verum
end;
hence Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = (((Partial_Product (Prob * (Complement C))) . k) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)) - ((((Prob * C) . (k + 1)) * ((Partial_Product (Prob * (Complement C))) . k)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q)) by A67, A66, PROB_1:33, XBOOLE_1:17; ::_thesis: verum
end;
(Prob * C) . (k + 1) = 1 - ((Prob * (Complement C)) . (k + 1))
proof
( C . (k + 1) = ((C . (k + 1)) `) ` & ((C . (k + 1)) `) ` = Omega \ ((C . (k + 1)) `) ) by SUBSET_1:def_4;
then ( Prob . (C . (k + 1)) = Prob . (([#] Sigma) \ ((C . (k + 1)) `)) & (C . (k + 1)) ` is Event of Sigma ) by PROB_1:20;
then A131: Prob . (C . (k + 1)) = 1 - (Prob . ((C . (k + 1)) `)) by PROB_1:32;
dom (Prob * C) = NAT by FUNCT_2:def_1;
then A132: (Prob * C) . (k + 1) = 1 - (Prob . ((C . (k + 1)) `)) by A131, FUNCT_1:12;
dom (Prob * (Complement C)) = NAT by FUNCT_2:def_1;
then (Prob * (Complement C)) . (k + 1) = Prob . ((Complement C) . (k + 1)) by FUNCT_1:12;
hence (Prob * C) . (k + 1) = 1 - ((Prob * (Complement C)) . (k + 1)) by A132, PROB_1:def_2; ::_thesis: verum
end;
then Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = (((Prob * (Complement C)) . (k + 1)) * ((Partial_Product (Prob * (Complement C))) . k)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) by A68;
hence Prob . (((Partial_Intersection (Complement C)) . (k + 1)) /\ ((Partial_Intersection (C ^\ (((k + 1) + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement C))) . (k + 1)) * ((Partial_Product (Prob * (C ^\ (((k + 1) + n2) + 1)))) . q) by SERIES_3:def_1; ::_thesis: verum
end;
A133: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A36, A61);
ex e being sequence of NAT st
( A * e = A & e is one-to-one & dom e <> {} )
proof
set e = Special_Function2 0;
A134: dom (Special_Function2 0) <> {} ;
( A is sequence of (bool Omega) & A * (Special_Function2 0) is sequence of (bool Omega) & ( for n being set st n in NAT holds
(A * (Special_Function2 0)) . n = A . n ) )
proof
A135: for n being set st n in NAT holds
( (A * (Special_Function2 0)) . n = A . n & A . ((Special_Function2 0) . n) = A . n )
proof
let n be set ; ::_thesis: ( n in NAT implies ( (A * (Special_Function2 0)) . n = A . n & A . ((Special_Function2 0) . n) = A . n ) )
assume n in NAT ; ::_thesis: ( (A * (Special_Function2 0)) . n = A . n & A . ((Special_Function2 0) . n) = A . n )
then reconsider n = n as Element of NAT ;
A136: (Special_Function2 0) . n = n + 0 by Def3;
dom (A * (Special_Function2 0)) = NAT by FUNCT_2:def_1;
hence ( (A * (Special_Function2 0)) . n = A . n & A . ((Special_Function2 0) . n) = A . n ) by A136, FUNCT_1:12; ::_thesis: verum
end;
thus ( A is sequence of (bool Omega) & A * (Special_Function2 0) is sequence of (bool Omega) & ( for n being set st n in NAT holds
(A * (Special_Function2 0)) . n = A . n ) ) by A135; ::_thesis: verum
end;
hence ex e being sequence of NAT st
( A * e = A & e is one-to-one & dom e <> {} ) by A134, FUNCT_2:12; ::_thesis: verum
end;
hence Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . q)) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . q) by A133; ::_thesis: verum
end;
(n - n1) - 1 is Element of NAT
proof
n1 + 1 <= n by A1, NAT_1:13;
then (n1 + 1) - 1 <= n - 1 by XREAL_1:9;
then ( n1 <= n - 1 & n - 1 is Element of NAT ) by A1, NAT_1:20;
then (n - 1) - n1 is Element of NAT by NAT_1:21;
hence (n - n1) - 1 is Element of NAT ; ::_thesis: verum
end;
hence Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) by A35; ::_thesis: verum
end;
theorem Th7: :: BOR_CANT:7
for Omega being non empty set
for Sigma being SigmaField of Omega
for A being SetSequence of Sigma
for n being Element of NAT holds (Partial_Intersection (Complement A)) . n = ((Partial_Union A) . n) `
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for A being SetSequence of Sigma
for n being Element of NAT holds (Partial_Intersection (Complement A)) . n = ((Partial_Union A) . n) `
let Sigma be SigmaField of Omega; ::_thesis: for A being SetSequence of Sigma
for n being Element of NAT holds (Partial_Intersection (Complement A)) . n = ((Partial_Union A) . n) `
let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds (Partial_Intersection (Complement A)) . n = ((Partial_Union A) . n) `
let n be Element of NAT ; ::_thesis: (Partial_Intersection (Complement A)) . n = ((Partial_Union A) . n) `
for x being set holds
( x in (Partial_Intersection (Complement A)) . n iff x in ((Partial_Union A) . n) ` )
proof
let x be set ; ::_thesis: ( x in (Partial_Intersection (Complement A)) . n iff x in ((Partial_Union A) . n) ` )
hereby ::_thesis: ( x in ((Partial_Union A) . n) ` implies x in (Partial_Intersection (Complement A)) . n )
assume A1: x in (Partial_Intersection (Complement A)) . n ; ::_thesis: x in ((Partial_Union A) . n) `
for knat being Nat st knat <= n holds
not x in A . knat
proof
let knat be Nat; ::_thesis: ( knat <= n implies not x in A . knat )
assume knat <= n ; ::_thesis: not x in A . knat
then A2: x in (Complement A) . knat by A1, PROB_3:25;
reconsider knat = knat as Element of NAT by ORDINAL1:def_12;
(Complement A) . knat = (A . knat) ` by PROB_1:def_2;
then (Complement A) . knat = Omega \ (A . knat) by SUBSET_1:def_4;
hence not x in A . knat by A2, XBOOLE_0:def_5; ::_thesis: verum
end;
then A3: not x in (Partial_Union A) . n by PROB_3:26;
x in Omega \ ((Partial_Union A) . n) by A1, A3, XBOOLE_0:def_5;
hence x in ((Partial_Union A) . n) ` by SUBSET_1:def_4; ::_thesis: verum
end;
assume A4: x in ((Partial_Union A) . n) ` ; ::_thesis: x in (Partial_Intersection (Complement A)) . n
x in Omega \ ((Partial_Union A) . n) by A4, SUBSET_1:def_4;
then A5: ( x in Omega & not x in (Partial_Union A) . n ) by XBOOLE_0:def_5;
for knat being Nat st knat <= n holds
x in (Complement A) . knat
proof
let knat be Nat; ::_thesis: ( knat <= n implies x in (Complement A) . knat )
assume knat <= n ; ::_thesis: x in (Complement A) . knat
then ( x in Omega & not x in A . knat ) by A5, PROB_3:26;
then A6: x in Omega \ (A . knat) by XBOOLE_0:def_5;
reconsider knat = knat as Element of NAT by ORDINAL1:def_12;
x in (A . knat) ` by A6, SUBSET_1:def_4;
hence x in (Complement A) . knat by PROB_1:def_2; ::_thesis: verum
end;
hence x in (Partial_Intersection (Complement A)) . n by PROB_3:25; ::_thesis: verum
end;
hence (Partial_Intersection (Complement A)) . n = ((Partial_Union A) . n) ` by TARSKI:1; ::_thesis: verum
end;
theorem Th8: :: BOR_CANT:8
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n))
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n))
let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n))
let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma
for n being Element of NAT holds Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n))
let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n))
let n be Element of NAT ; ::_thesis: Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n))
A1: Prob . ((Partial_Intersection (Complement A)) . n) = Prob . (((Partial_Union A) . n) `) by Th7;
Prob . (((Partial_Union A) . n) `) = Prob . (([#] Sigma) \ ((Partial_Union A) . n)) by SUBSET_1:def_4;
hence Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n)) by A1, PROB_1:32; ::_thesis: verum
end;
definition
let X be set ;
let A be SetSequence of X;
func Union_Shift_Seq A -> SetSequence of X means :Def7: :: BOR_CANT:def 7
for n being Element of NAT holds it . n = Union (A ^\ n);
existence
ex b1 being SetSequence of X st
for n being Element of NAT holds b1 . n = Union (A ^\ n)
proof
for X being set
for A being SetSequence of X ex S being SetSequence of X st
for n being Element of NAT holds S . n = Union (A ^\ n)
proof
let X be set ; ::_thesis: for A being SetSequence of X ex S being SetSequence of X st
for n being Element of NAT holds S . n = Union (A ^\ n)
let A be SetSequence of X; ::_thesis: ex S being SetSequence of X st
for n being Element of NAT holds S . n = Union (A ^\ n)
ex J being SetSequence of X st
( J . 0 = Union (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Union (A ^\ (n + 1)) ) )
proof
defpred S1[ set , set , set ] means for x, y being Subset of X
for k being Element of NAT st k = $1 & x = $2 & y = $3 holds
y = Union (A ^\ (k + 1));
A1: for n being Element of NAT
for x being Subset of X ex y being Subset of X st S1[n,x,y]
proof
let n be Element of NAT ; ::_thesis: for x being Subset of X ex y being Subset of X st S1[n,x,y]
let x be Subset of X; ::_thesis: ex y being Subset of X st S1[n,x,y]
take y = Union (A ^\ (n + 1)); ::_thesis: S1[n,x,y]
thus S1[n,x,y] ; ::_thesis: verum
end;
consider J being SetSequence of X such that
A2: J . 0 = Union (A ^\ 0) and
A3: for n being Element of NAT holds S1[n,J . n,J . (n + 1)] from RECDEF_1:sch_2(A1);
take J ; ::_thesis: ( J . 0 = Union (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Union (A ^\ (n + 1)) ) )
thus J . 0 = Union (A ^\ 0) by A2; ::_thesis: for n being Element of NAT holds J . (n + 1) = Union (A ^\ (n + 1))
let n be Element of NAT ; ::_thesis: J . (n + 1) = Union (A ^\ (n + 1))
S1[n,J . n,J . (n + 1)] by A3;
hence J . (n + 1) = Union (A ^\ (n + 1)) ; ::_thesis: verum
end;
then consider J being SetSequence of X such that
A4: ( J . 0 = Union (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Union (A ^\ (n + 1)) ) ) ;
A5: ( J . 0 = Union (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Union (A ^\ (n + 1)) ) implies for n being Element of NAT holds J . n = Union (A ^\ n) )
proof
assume A6: ( J . 0 = Union (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Union (A ^\ (n + 1)) ) ) ; ::_thesis: for n being Element of NAT holds J . n = Union (A ^\ n)
let n be Nat; ::_thesis: ( n is Element of REAL & n is Element of NAT implies J . n = Union (A ^\ n) )
percases ( n = 0 or ex q being Nat st n = q + 1 ) by NAT_1:6;
suppose n = 0 ; ::_thesis: ( n is Element of REAL & n is Element of NAT implies J . n = Union (A ^\ n) )
hence ( n is Element of REAL & n is Element of NAT implies J . n = Union (A ^\ n) ) by A6; ::_thesis: verum
end;
suppose ex q being Nat st n = q + 1 ; ::_thesis: ( n is Element of REAL & n is Element of NAT implies J . n = Union (A ^\ n) )
then consider q being Nat such that
A7: n = q + 1 ;
reconsider q = q as Element of NAT by ORDINAL1:def_12;
J . (q + 1) = Union (A ^\ (q + 1)) by A6;
hence ( n is Element of REAL & n is Element of NAT implies J . n = Union (A ^\ n) ) by A7; ::_thesis: verum
end;
end;
end;
take J ; ::_thesis: for n being Element of NAT holds J . n = Union (A ^\ n)
thus for n being Element of NAT holds J . n = Union (A ^\ n) by A4, A5; ::_thesis: verum
end;
hence ex b1 being SetSequence of X st
for n being Element of NAT holds b1 . n = Union (A ^\ n) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being SetSequence of X st ( for n being Element of NAT holds b1 . n = Union (A ^\ n) ) & ( for n being Element of NAT holds b2 . n = Union (A ^\ n) ) holds
b1 = b2
proof
let J1, J2 be SetSequence of X; ::_thesis: ( ( for n being Element of NAT holds J1 . n = Union (A ^\ n) ) & ( for n being Element of NAT holds J2 . n = Union (A ^\ n) ) implies J1 = J2 )
assume that
A8: for n being Element of NAT holds J1 . n = Union (A ^\ n) and
A9: for n being Element of NAT holds J2 . n = Union (A ^\ n) ; ::_thesis: J1 = J2
for n being Element of NAT holds J1 . n = J2 . n
proof
let n be Element of NAT ; ::_thesis: J1 . n = J2 . n
J1 . n = Union (A ^\ n) by A8;
hence J1 . n = J2 . n by A9; ::_thesis: verum
end;
then for n being set st n in NAT holds
J1 . n = J2 . n ;
hence J1 = J2 by FUNCT_2:12; ::_thesis: verum
end;
end;
:: deftheorem Def7 defines Union_Shift_Seq BOR_CANT:def_7_:_
for X being set
for A, b3 being SetSequence of X holds
( b3 = Union_Shift_Seq A iff for n being Element of NAT holds b3 . n = Union (A ^\ n) );
registration
let Omega be non empty set ;
let Sigma be SigmaField of Omega;
let A be SetSequence of Sigma;
cluster Union_Shift_Seq A -> Sigma -valued ;
coherence
Union_Shift_Seq A is Sigma -valued
proof
defpred S1[ set ] means (Union_Shift_Seq A) . Omega is Event of Sigma;
(Union_Shift_Seq A) . 0 = Union (A ^\ 0) by Def7;
then A1: S1[ 0 ] by PROB_1:17;
A2: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume (Union_Shift_Seq A) . k is Event of Sigma ; ::_thesis: S1[k + 1]
Union (A ^\ (k + 1)) in Sigma by PROB_1:17;
hence S1[k + 1] by Def7; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A1, A2);
hence Union_Shift_Seq A is Sigma -valued by PROB_1:25; ::_thesis: verum
end;
end;
definition
let Omega be non empty set ;
let Sigma be SigmaField of Omega;
let A be SetSequence of Sigma;
func @lim_sup A -> Event of Sigma equals :: BOR_CANT:def 8
@Intersection (Union_Shift_Seq A);
correctness
coherence
@Intersection (Union_Shift_Seq A) is Event of Sigma;
;
end;
:: deftheorem defines @lim_sup BOR_CANT:def_8_:_
for Omega being non empty set
for Sigma being SigmaField of Omega
for A being SetSequence of Sigma holds @lim_sup A = @Intersection (Union_Shift_Seq A);
definition
let X be set ;
let A be SetSequence of X;
func Intersect_Shift_Seq A -> SetSequence of X means :Def9: :: BOR_CANT:def 9
for n being Element of NAT holds it . n = Intersection (A ^\ n);
existence
ex b1 being SetSequence of X st
for n being Element of NAT holds b1 . n = Intersection (A ^\ n)
proof
for X being set
for A being SetSequence of X ex S being SetSequence of X st
for n being Element of NAT holds S . n = Intersection (A ^\ n)
proof
let X be set ; ::_thesis: for A being SetSequence of X ex S being SetSequence of X st
for n being Element of NAT holds S . n = Intersection (A ^\ n)
let A be SetSequence of X; ::_thesis: ex S being SetSequence of X st
for n being Element of NAT holds S . n = Intersection (A ^\ n)
A1: ex J being SetSequence of X st
( J . 0 = Intersection (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Intersection (A ^\ (n + 1)) ) )
proof
defpred S1[ set , set , set ] means for x, y being Subset of X
for k being Element of NAT st k = $1 & x = $2 & y = $3 holds
y = Intersection (A ^\ (k + 1));
A2: for n being Element of NAT
for x being Subset of X ex y being Subset of X st S1[n,x,y]
proof
let n be Element of NAT ; ::_thesis: for x being Subset of X ex y being Subset of X st S1[n,x,y]
let x be Subset of X; ::_thesis: ex y being Subset of X st S1[n,x,y]
take y = Intersection (A ^\ (n + 1)); ::_thesis: S1[n,x,y]
thus S1[n,x,y] ; ::_thesis: verum
end;
consider J being SetSequence of X such that
A3: J . 0 = Intersection (A ^\ 0) and
A4: for n being Element of NAT holds S1[n,J . n,J . (n + 1)] from RECDEF_1:sch_2(A2);
take J ; ::_thesis: ( J . 0 = Intersection (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Intersection (A ^\ (n + 1)) ) )
thus J . 0 = Intersection (A ^\ 0) by A3; ::_thesis: for n being Element of NAT holds J . (n + 1) = Intersection (A ^\ (n + 1))
let n be Element of NAT ; ::_thesis: J . (n + 1) = Intersection (A ^\ (n + 1))
S1[n,J . n,J . (n + 1)] by A4;
hence J . (n + 1) = Intersection (A ^\ (n + 1)) ; ::_thesis: verum
end;
consider J being SetSequence of X such that
A5: ( J . 0 = Intersection (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Intersection (A ^\ (n + 1)) ) ) by A1;
A6: ( J . 0 = Intersection (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Intersection (A ^\ (n + 1)) ) implies for n being Element of NAT holds J . n = Intersection (A ^\ n) )
proof
assume A7: ( J . 0 = Intersection (A ^\ 0) & ( for n being Element of NAT holds J . (n + 1) = Intersection (A ^\ (n + 1)) ) ) ; ::_thesis: for n being Element of NAT holds J . n = Intersection (A ^\ n)
let n be Nat; ::_thesis: ( n is Element of REAL & n is Element of NAT implies J . n = Intersection (A ^\ n) )
percases ( n = 0 or ex q being Nat st n = q + 1 ) by NAT_1:6;
suppose n = 0 ; ::_thesis: ( n is Element of REAL & n is Element of NAT implies J . n = Intersection (A ^\ n) )
hence ( n is Element of REAL & n is Element of NAT implies J . n = Intersection (A ^\ n) ) by A7; ::_thesis: verum
end;
suppose ex q being Nat st n = q + 1 ; ::_thesis: ( n is Element of REAL & n is Element of NAT implies J . n = Intersection (A ^\ n) )
then consider q being Nat such that
A8: n = q + 1 ;
reconsider q = q as Element of NAT by ORDINAL1:def_12;
J . (q + 1) = Intersection (A ^\ (q + 1)) by A7;
hence ( n is Element of REAL & n is Element of NAT implies J . n = Intersection (A ^\ n) ) by A8; ::_thesis: verum
end;
end;
end;
take J ; ::_thesis: for n being Element of NAT holds J . n = Intersection (A ^\ n)
thus for n being Element of NAT holds J . n = Intersection (A ^\ n) by A5, A6; ::_thesis: verum
end;
hence ex b1 being SetSequence of X st
for n being Element of NAT holds b1 . n = Intersection (A ^\ n) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being SetSequence of X st ( for n being Element of NAT holds b1 . n = Intersection (A ^\ n) ) & ( for n being Element of NAT holds b2 . n = Intersection (A ^\ n) ) holds
b1 = b2
proof
let J1, J2 be SetSequence of X; ::_thesis: ( ( for n being Element of NAT holds J1 . n = Intersection (A ^\ n) ) & ( for n being Element of NAT holds J2 . n = Intersection (A ^\ n) ) implies J1 = J2 )
assume that
A9: for n being Element of NAT holds J1 . n = Intersection (A ^\ n) and
A10: for n being Element of NAT holds J2 . n = Intersection (A ^\ n) ; ::_thesis: J1 = J2
for n being Element of NAT holds J1 . n = J2 . n
proof
let n be Element of NAT ; ::_thesis: J1 . n = J2 . n
J1 . n = Intersection (A ^\ n) by A9;
hence J1 . n = J2 . n by A10; ::_thesis: verum
end;
then for n being set st n in NAT holds
J1 . n = J2 . n ;
hence J1 = J2 by FUNCT_2:12; ::_thesis: verum
end;
end;
:: deftheorem Def9 defines Intersect_Shift_Seq BOR_CANT:def_9_:_
for X being set
for A, b3 being SetSequence of X holds
( b3 = Intersect_Shift_Seq A iff for n being Element of NAT holds b3 . n = Intersection (A ^\ n) );
registration
let Omega be non empty set ;
let Sigma be SigmaField of Omega;
let A be SetSequence of Sigma;
cluster Intersect_Shift_Seq A -> Sigma -valued ;
coherence
Intersect_Shift_Seq A is Sigma -valued
proof
defpred S1[ set ] means (Intersect_Shift_Seq A) . Omega is Event of Sigma;
A1: Union (Complement (A ^\ 0)) is Event of Sigma by PROB_1:26;
(Intersect_Shift_Seq A) . 0 = Intersection (A ^\ 0) by Def9;
then A2: S1[ 0 ] by A1, PROB_1:20;
A3: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume (Intersect_Shift_Seq A) . k is Event of Sigma ; ::_thesis: S1[k + 1]
A4: Union (Complement (A ^\ (k + 1))) is Event of Sigma by PROB_1:26;
(Intersect_Shift_Seq A) . (k + 1) = Intersection (A ^\ (k + 1)) by Def9;
hence S1[k + 1] by A4, PROB_1:20; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A2, A3);
hence Intersect_Shift_Seq A is Sigma -valued by PROB_1:25; ::_thesis: verum
end;
end;
definition
let Omega be non empty set ;
let Sigma be SigmaField of Omega;
let A be SetSequence of Sigma;
func @lim_inf A -> Event of Sigma equals :: BOR_CANT:def 10
Union (Intersect_Shift_Seq A);
correctness
coherence
Union (Intersect_Shift_Seq A) is Event of Sigma;
by PROB_1:26;
end;
:: deftheorem defines @lim_inf BOR_CANT:def_10_:_
for Omega being non empty set
for Sigma being SigmaField of Omega
for A being SetSequence of Sigma holds @lim_inf A = Union (Intersect_Shift_Seq A);
theorem Th9: :: BOR_CANT:9
for Omega being non empty set
for Sigma being SigmaField of Omega
for A being SetSequence of Sigma
for n being Element of NAT holds (Intersect_Shift_Seq (Complement A)) . n = ((Union_Shift_Seq A) . n) `
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for A being SetSequence of Sigma
for n being Element of NAT holds (Intersect_Shift_Seq (Complement A)) . n = ((Union_Shift_Seq A) . n) `
let Sigma be SigmaField of Omega; ::_thesis: for A being SetSequence of Sigma
for n being Element of NAT holds (Intersect_Shift_Seq (Complement A)) . n = ((Union_Shift_Seq A) . n) `
let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds (Intersect_Shift_Seq (Complement A)) . n = ((Union_Shift_Seq A) . n) `
let n be Element of NAT ; ::_thesis: (Intersect_Shift_Seq (Complement A)) . n = ((Union_Shift_Seq A) . n) `
for x being set holds
( x in (Intersect_Shift_Seq (Complement A)) . n iff x in ((Union_Shift_Seq A) . n) ` )
proof
let x be set ; ::_thesis: ( x in (Intersect_Shift_Seq (Complement A)) . n iff x in ((Union_Shift_Seq A) . n) ` )
hereby ::_thesis: ( x in ((Union_Shift_Seq A) . n) ` implies x in (Intersect_Shift_Seq (Complement A)) . n )
assume A1: x in (Intersect_Shift_Seq (Complement A)) . n ; ::_thesis: x in ((Union_Shift_Seq A) . n) `
then A2: x in Intersection ((Complement A) ^\ n) by Def9;
A3: for k being Element of NAT holds not x in (A ^\ n) . k
proof
let k be Element of NAT ; ::_thesis: not x in (A ^\ n) . k
x in ((Complement A) ^\ n) . k by A2, PROB_1:13;
then x in (Complement A) . (n + k) by NAT_1:def_3;
then A4: x in (A . (n + k)) ` by PROB_1:def_2;
x in Omega \ (A . (n + k)) by A4, SUBSET_1:def_4;
then ( x in Omega & not x in A . (n + k) ) by XBOOLE_0:def_5;
hence not x in (A ^\ n) . k by NAT_1:def_3; ::_thesis: verum
end;
A5: not x in Union (A ^\ n) by A3, PROB_1:12;
A6: not x in (Union_Shift_Seq A) . n by Def7, A5;
A7: x in Omega \ ((Union_Shift_Seq A) . n) by A1, A6, XBOOLE_0:def_5;
thus x in ((Union_Shift_Seq A) . n) ` by A7, SUBSET_1:def_4; ::_thesis: verum
end;
assume A8: x in ((Union_Shift_Seq A) . n) ` ; ::_thesis: x in (Intersect_Shift_Seq (Complement A)) . n
A9: ( x in ((Union_Shift_Seq A) . n) ` iff x in Omega \ ((Union_Shift_Seq A) . n) ) by SUBSET_1:def_4;
A10: ( x in (Union_Shift_Seq A) . n iff x in Union (A ^\ n) ) by Def7;
A11: for k being Element of NAT holds x in ((Complement A) ^\ n) . k
proof
let k be Element of NAT ; ::_thesis: x in ((Complement A) ^\ n) . k
A12: not x in (A ^\ n) . k by A10, A8, A9, PROB_1:12, XBOOLE_0:def_5;
A13: not x in A . (n + k) by A12, NAT_1:def_3;
A14: x in Omega \ (A . (n + k)) by A8, A13, XBOOLE_0:def_5;
( x in (A . (n + k)) ` iff x in (Complement A) . (n + k) ) by PROB_1:def_2;
hence x in ((Complement A) ^\ n) . k by A14, NAT_1:def_3, SUBSET_1:def_4; ::_thesis: verum
end;
x in Intersection ((Complement A) ^\ n) by A11, PROB_1:13;
hence x in (Intersect_Shift_Seq (Complement A)) . n by Def9; ::_thesis: verum
end;
hence (Intersect_Shift_Seq (Complement A)) . n = ((Union_Shift_Seq A) . n) ` by TARSKI:1; ::_thesis: verum
end;
theorem Th10: :: BOR_CANT:10
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT st A is_all_independent_wrt Prob holds
Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT st A is_all_independent_wrt Prob holds
Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n
let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT st A is_all_independent_wrt Prob holds
Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n
let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma
for n being Element of NAT st A is_all_independent_wrt Prob holds
Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n
let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT st A is_all_independent_wrt Prob holds
Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n
let n be Element of NAT ; ::_thesis: ( A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n )
assume A1: A is_all_independent_wrt Prob ; ::_thesis: Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n
defpred S1[ Element of NAT ] means Prob . ((Partial_Intersection (Complement A)) . $1) = (Partial_Product (Prob * (Complement A))) . $1;
dom (Prob * (Complement A)) = NAT by FUNCT_2:def_1;
then A2: (Prob * (Complement A)) . 0 = Prob . ((Complement A) . 0) by FUNCT_1:12;
A3: (Partial_Product (Prob * (Complement A))) . 0 = (Prob * (Complement A)) . 0 by SERIES_3:def_1;
A4: S1[ 0 ] by A2, A3, PROB_3:21;
A5: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A6: S1[k] ; ::_thesis: S1[k + 1]
(((Partial_Intersection (Complement A)) . k) /\ ((Partial_Intersection (Complement A)) . k)) /\ ((Complement A) . (k + 1)) = ((Partial_Intersection (Complement A)) . k) /\ ((A . (k + 1)) `) by PROB_1:def_2;
then (((Partial_Intersection (Complement A)) . k) /\ ((Partial_Intersection (Complement A)) . k)) /\ ((Complement A) . (k + 1)) = ((Partial_Intersection (Complement A)) . k) /\ (Omega \ (A . (k + 1))) by SUBSET_1:def_4;
then A7: (((Partial_Intersection (Complement A)) . k) /\ ((Partial_Intersection (Complement A)) . k)) /\ ((Complement A) . (k + 1)) = (((Partial_Intersection (Complement A)) . k) /\ Omega) \ (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) by XBOOLE_1:50;
A8: ((Partial_Intersection (Complement A)) . k) /\ Omega = (Partial_Intersection (Complement A)) . k by XBOOLE_1:28;
A9: Prob . (((Partial_Intersection (Complement A)) . k) \ (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1)))) = (Prob . ((Partial_Intersection (Complement A)) . k)) - (Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1)))) by PROB_1:33, XBOOLE_1:17;
A10: Prob . ((Partial_Intersection (Complement A)) . (k + 1)) = (Prob . ((Partial_Intersection (Complement A)) . k)) - (Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1)))) by A7, A8, A9, PROB_3:21;
for A being SetSequence of Sigma
for k being Element of NAT st A is_all_independent_wrt Prob holds
Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) = ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * A) . (k + 1))
proof
let A be SetSequence of Sigma; ::_thesis: for k being Element of NAT st A is_all_independent_wrt Prob holds
Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) = ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * A) . (k + 1))
let k be Element of NAT ; ::_thesis: ( A is_all_independent_wrt Prob implies Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) = ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * A) . (k + 1)) )
assume A11: A is_all_independent_wrt Prob ; ::_thesis: Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) = ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * A) . (k + 1))
consider n being Element of NAT such that
A12: n = k + 1 ;
consider n1 being Element of NAT such that
A13: n1 = k ;
n1 < k + 1 by A13, NAT_1:13;
then Prob . (((Partial_Intersection (Complement A)) . k) /\ ((Partial_Intersection (A ^\ ((k + 0) + 1))) . ((n - k) - 1))) = ((Partial_Product (Prob * (Complement A))) . k) * ((Partial_Product (Prob * (A ^\ ((k + 0) + 1)))) . ((n - k) - 1)) by A12, A11, Th6, A13;
then A14: Prob . (((Partial_Intersection (Complement A)) . k) /\ ((A ^\ (k + 1)) . 0)) = ((Partial_Product (Prob * (Complement A))) . k) * ((Partial_Product (Prob * (A ^\ (k + 1)))) . 0) by A12, PROB_3:21;
A15: (A ^\ (k + 1)) . 0 = A . (0 + (k + 1)) by NAT_1:def_3;
then A16: Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) = ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * (A ^\ (k + 1))) . 0) by A14, SERIES_3:def_1;
dom (Prob * (A ^\ (k + 1))) = NAT by FUNCT_2:def_1;
then A17: Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) = ((Partial_Product (Prob * (Complement A))) . k) * (Prob . (A . (k + 1))) by A15, A16, FUNCT_1:12;
dom (Prob * A) = NAT by FUNCT_2:def_1;
hence Prob . (((Partial_Intersection (Complement A)) . k) /\ (A . (k + 1))) = ((Partial_Product (Prob * (Complement A))) . k) * ((Prob * A) . (k + 1)) by A17, FUNCT_1:12; ::_thesis: verum
end;
then A18: Prob . ((Partial_Intersection (Complement A)) . (k + 1)) = ((Partial_Product (Prob * (Complement A))) . k) - (((Partial_Product (Prob * (Complement A))) . k) * ((Prob * A) . (k + 1))) by A6, A10, A1;
( A . (k + 1) = ((A . (k + 1)) `) ` & ((A . (k + 1)) `) ` = Omega \ ((A . (k + 1)) `) ) by SUBSET_1:def_4;
then ( Prob . (A . (k + 1)) = Prob . (([#] Sigma) \ ((A . (k + 1)) `)) & (A . (k + 1)) ` is Event of Sigma ) by PROB_1:20;
then A19: Prob . (A . (k + 1)) = 1 - (Prob . ((A . (k + 1)) `)) by PROB_1:32;
dom (Prob * A) = NAT by FUNCT_2:def_1;
then A20: (Prob * A) . (k + 1) = 1 - (Prob . ((A . (k + 1)) `)) by A19, FUNCT_1:12;
dom (Prob * (Complement A)) = NAT by FUNCT_2:def_1;
then A21: (Prob * (Complement A)) . (k + 1) = Prob . ((Complement A) . (k + 1)) by FUNCT_1:12;
(Prob * A) . (k + 1) = 1 - ((Prob * (Complement A)) . (k + 1)) by A21, A20, PROB_1:def_2;
then Prob . ((Partial_Intersection (Complement A)) . (k + 1)) = (((Partial_Product (Prob * (Complement A))) . k) - ((Partial_Product (Prob * (Complement A))) . k)) + (((Partial_Product (Prob * (Complement A))) . k) * ((Prob * (Complement A)) . (k + 1))) by A18;
hence S1[k + 1] by SERIES_3:def_1; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A4, A5);
hence Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n ; ::_thesis: verum
end;
theorem Th11: :: BOR_CANT:11
for X being set
for A being SetSequence of X holds
( superior_setsequence A = Union_Shift_Seq A & inferior_setsequence A = Intersect_Shift_Seq A )
proof
let X be set ; ::_thesis: for A being SetSequence of X holds
( superior_setsequence A = Union_Shift_Seq A & inferior_setsequence A = Intersect_Shift_Seq A )
let A be SetSequence of X; ::_thesis: ( superior_setsequence A = Union_Shift_Seq A & inferior_setsequence A = Intersect_Shift_Seq A )
thus superior_setsequence A = Union_Shift_Seq A ::_thesis: inferior_setsequence A = Intersect_Shift_Seq A
proof
let n be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: (superior_setsequence A) . n = (Union_Shift_Seq A) . n
for x being set holds
( x in (superior_setsequence A) . n iff x in (Union_Shift_Seq A) . n )
proof
let x be set ; ::_thesis: ( x in (superior_setsequence A) . n iff x in (Union_Shift_Seq A) . n )
hereby ::_thesis: ( x in (Union_Shift_Seq A) . n implies x in (superior_setsequence A) . n )
assume x in (superior_setsequence A) . n ; ::_thesis: x in (Union_Shift_Seq A) . n
then consider k being Element of NAT such that
A1: x in A . (n + k) by SETLIM_1:20;
x in (A ^\ n) . k by A1, NAT_1:def_3;
then x in Union (A ^\ n) by PROB_1:12;
hence x in (Union_Shift_Seq A) . n by Def7; ::_thesis: verum
end;
assume x in (Union_Shift_Seq A) . n ; ::_thesis: x in (superior_setsequence A) . n
then x in Union (A ^\ n) by Def7;
then consider k being Element of NAT such that
A2: x in (A ^\ n) . k by PROB_1:12;
x in A . (n + k) by A2, NAT_1:def_3;
hence x in (superior_setsequence A) . n by SETLIM_1:20; ::_thesis: verum
end;
hence (superior_setsequence A) . n = (Union_Shift_Seq A) . n by TARSKI:1; ::_thesis: verum
end;
let n be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: (inferior_setsequence A) . n = (Intersect_Shift_Seq A) . n
for x being set holds
( x in (inferior_setsequence A) . n iff x in (Intersect_Shift_Seq A) . n )
proof
let x be set ; ::_thesis: ( x in (inferior_setsequence A) . n iff x in (Intersect_Shift_Seq A) . n )
hereby ::_thesis: ( x in (Intersect_Shift_Seq A) . n implies x in (inferior_setsequence A) . n )
assume A3: x in (inferior_setsequence A) . n ; ::_thesis: x in (Intersect_Shift_Seq A) . n
A4: for k being Element of NAT holds x in (A ^\ n) . k
proof
let k be Element of NAT ; ::_thesis: x in (A ^\ n) . k
x in A . (k + n) by A3, SETLIM_1:19;
hence x in (A ^\ n) . k by NAT_1:def_3; ::_thesis: verum
end;
x in Intersection (A ^\ n) by A4, PROB_1:13;
hence x in (Intersect_Shift_Seq A) . n by Def9; ::_thesis: verum
end;
assume x in (Intersect_Shift_Seq A) . n ; ::_thesis: x in (inferior_setsequence A) . n
then A5: x in Intersection (A ^\ n) by Def9;
for k being Element of NAT holds x in A . (n + k)
proof
let k be Element of NAT ; ::_thesis: x in A . (n + k)
x in (A ^\ n) . k by A5, PROB_1:13;
hence x in A . (n + k) by NAT_1:def_3; ::_thesis: verum
end;
hence x in (inferior_setsequence A) . n by SETLIM_1:19; ::_thesis: verum
end;
hence (inferior_setsequence A) . n = (Intersect_Shift_Seq A) . n by TARSKI:1; ::_thesis: verum
end;
theorem :: BOR_CANT:12
for Omega being non empty set
for Sigma being SigmaField of Omega
for A being SetSequence of Sigma holds
( superior_setsequence A = Union_Shift_Seq A & inferior_setsequence A = Intersect_Shift_Seq A ) by Th11;
definition
let Omega be non empty set ;
let Sigma be SigmaField of Omega;
let Prob be Probability of Sigma;
let A be SetSequence of Sigma;
func Sum_Shift_Seq (Prob,A) -> Real_Sequence means :Def11: :: BOR_CANT:def 11
for n being Element of NAT holds it . n = Sum (Prob * (A ^\ n));
existence
ex b1 being Real_Sequence st
for n being Element of NAT holds b1 . n = Sum (Prob * (A ^\ n))
proof
deffunc H1( Element of NAT ) -> Element of REAL = Sum (Prob * (A ^\ $1));
consider f being Real_Sequence such that
A1: for k being Element of NAT holds f . k = H1(k) from FUNCT_2:sch_4();
take f ; ::_thesis: for n being Element of NAT holds f . n = Sum (Prob * (A ^\ n))
let knat be Nat; ::_thesis: ( knat is Element of REAL & knat is Element of NAT implies f . knat = Sum (Prob * (A ^\ knat)) )
thus ( knat is Element of REAL & knat is Element of NAT implies f . knat = Sum (Prob * (A ^\ knat)) ) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being Real_Sequence st ( for n being Element of NAT holds b1 . n = Sum (Prob * (A ^\ n)) ) & ( for n being Element of NAT holds b2 . n = Sum (Prob * (A ^\ n)) ) holds
b1 = b2
proof
let J1, J2 be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds J1 . n = Sum (Prob * (A ^\ n)) ) & ( for n being Element of NAT holds J2 . n = Sum (Prob * (A ^\ n)) ) implies J1 = J2 )
assume A2: for n being Element of NAT holds J1 . n = Sum (Prob * (A ^\ n)) ; ::_thesis: ( ex n being Element of NAT st not J2 . n = Sum (Prob * (A ^\ n)) or J1 = J2 )
assume A3: for n being Element of NAT holds J2 . n = Sum (Prob * (A ^\ n)) ; ::_thesis: J1 = J2
let n be Element of NAT ; :: according to FUNCT_2:def_8 ::_thesis: J1 . n = J2 . n
J1 . n = Sum (Prob * (A ^\ n)) by A2;
hence J1 . n = J2 . n by A3; ::_thesis: verum
end;
end;
:: deftheorem Def11 defines Sum_Shift_Seq BOR_CANT:def_11_:_
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for b5 being Real_Sequence holds
( b5 = Sum_Shift_Seq (Prob,A) iff for n being Element of NAT holds b5 . n = Sum (Prob * (A ^\ n)) );
theorem Th13: :: BOR_CANT:13
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds
( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent )
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds
( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent )
let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma
for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds
( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent )
let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds
( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent )
let A be SetSequence of Sigma; ::_thesis: ( Partial_Sums (Prob * A) is convergent implies ( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) )
assume A1: Partial_Sums (Prob * A) is convergent ; ::_thesis: ( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent )
A2: Prob * A is summable by A1, SERIES_1:def_2;
A3: for n being Element of NAT holds 0 <= (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n
proof
let n be Element of NAT ; ::_thesis: 0 <= (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n
A4: dom (Prob * (Partial_Intersection (Union_Shift_Seq A))) = NAT by FUNCT_2:def_1;
(Prob * (Partial_Intersection (Union_Shift_Seq A))) . n = Prob . ((Partial_Intersection (Union_Shift_Seq A)) . n) by A4, FUNCT_1:12;
hence 0 <= (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n by PROB_1:def_8; ::_thesis: verum
end;
A5: Intersection (Partial_Intersection (Union_Shift_Seq A)) = Intersection (Union_Shift_Seq A) by PROB_3:29;
A6: Partial_Intersection (Union_Shift_Seq A) is non-ascending by PROB_3:27;
A7: ( lim (Prob * (Partial_Intersection (Union_Shift_Seq A))) = Prob . (Intersection (Partial_Intersection (Union_Shift_Seq A))) & Prob * (Partial_Intersection (Union_Shift_Seq A)) is convergent ) by A6, PROB_1:def_8;
A8: for A being SetSequence of Sigma
for n, s being Element of NAT holds (Prob * (Partial_Union (A ^\ s))) . n <= (Partial_Sums (Prob * (A ^\ s))) . n
proof
let A be SetSequence of Sigma; ::_thesis: for n, s being Element of NAT holds (Prob * (Partial_Union (A ^\ s))) . n <= (Partial_Sums (Prob * (A ^\ s))) . n
let n, s be Element of NAT ; ::_thesis: (Prob * (Partial_Union (A ^\ s))) . n <= (Partial_Sums (Prob * (A ^\ s))) . n
defpred S1[ set ] means (Prob * (Partial_Union (A ^\ s))) . $1 <= (Partial_Sums (Prob * (A ^\ s))) . $1;
A9: (Partial_Sums (Prob * (A ^\ s))) . 0 = (Prob * (A ^\ s)) . 0 by SERIES_1:def_1;
A10: dom (Prob * (A ^\ s)) = NAT by FUNCT_2:def_1;
A11: (Prob * (A ^\ s)) . 0 = Prob . ((A ^\ s) . 0) by A10, FUNCT_1:12;
A12: Prob . ((Partial_Union (A ^\ s)) . 0) = Prob . ((A ^\ s) . 0) by PROB_3:def_2;
A13: dom (Prob * (Partial_Union (A ^\ s))) = NAT by FUNCT_2:def_1;
A14: S1[ 0 ] by A13, A12, A11, A9, FUNCT_1:12;
A15: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A16: (Prob * (Partial_Union (A ^\ s))) . k <= (Partial_Sums (Prob * (A ^\ s))) . k ; ::_thesis: S1[k + 1]
A17: dom (Prob * (Partial_Union (A ^\ s))) = NAT by FUNCT_2:def_1;
A18: Prob . (((Partial_Union (A ^\ s)) . k) \/ ((A ^\ s) . (k + 1))) <= (Prob . ((Partial_Union (A ^\ s)) . k)) + (Prob . ((A ^\ s) . (k + 1))) by PROB_1:39;
dom (Prob * (A ^\ s)) = NAT by FUNCT_2:def_1;
then A19: (Prob * (A ^\ s)) . (k + 1) = Prob . ((A ^\ s) . (k + 1)) by FUNCT_1:12;
A20: ( Prob . ((Partial_Union (A ^\ s)) . (k + 1)) <= (Prob . ((Partial_Union (A ^\ s)) . k)) + ((Prob * (A ^\ s)) . (k + 1)) implies (Prob . ((Partial_Union (A ^\ s)) . (k + 1))) - (Prob . ((Partial_Union (A ^\ s)) . k)) <= (Prob * (A ^\ s)) . (k + 1) ) by XREAL_1:20;
A21: ( (Prob . ((Partial_Union (A ^\ s)) . (k + 1))) - ((Prob * (A ^\ s)) . (k + 1)) <= Prob . ((Partial_Union (A ^\ s)) . k) & Prob . ((Partial_Union (A ^\ s)) . k) <= (Partial_Sums (Prob * (A ^\ s))) . k implies (Prob . ((Partial_Union (A ^\ s)) . (k + 1))) - ((Prob * (A ^\ s)) . (k + 1)) <= (Partial_Sums (Prob * (A ^\ s))) . k ) by XXREAL_0:2;
A22: ( (Prob . ((Partial_Union (A ^\ s)) . (k + 1))) - ((Prob * (A ^\ s)) . (k + 1)) <= (Partial_Sums (Prob * (A ^\ s))) . k implies Prob . ((Partial_Union (A ^\ s)) . (k + 1)) <= ((Partial_Sums (Prob * (A ^\ s))) . k) + ((Prob * (A ^\ s)) . (k + 1)) ) by XREAL_1:20;
A23: Prob . ((Partial_Union (A ^\ s)) . (k + 1)) <= (Partial_Sums (Prob * (A ^\ s))) . (k + 1) by A18, A19, A20, A17, A16, A21, A22, FUNCT_1:12, PROB_3:def_2, SERIES_1:def_1, XREAL_1:12;
dom (Prob * (Partial_Union (A ^\ s))) = NAT by FUNCT_2:def_1;
hence S1[k + 1] by A23, FUNCT_1:12; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A14, A15);
hence (Prob * (Partial_Union (A ^\ s))) . n <= (Partial_Sums (Prob * (A ^\ s))) . n ; ::_thesis: verum
end;
A24: for k being Element of NAT holds Partial_Sums ((Prob * A) ^\ k) is convergent
proof
let k be Element of NAT ; ::_thesis: Partial_Sums ((Prob * A) ^\ k) is convergent
(Prob * A) ^\ k is summable by A2, SERIES_1:12;
hence Partial_Sums ((Prob * A) ^\ k) is convergent by SERIES_1:def_2; ::_thesis: verum
end;
A25: for A being SetSequence of Sigma
for n being Element of NAT holds Prob * (A ^\ n) = (Prob * A) ^\ n
proof
let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds Prob * (A ^\ n) = (Prob * A) ^\ n
let n be Element of NAT ; ::_thesis: Prob * (A ^\ n) = (Prob * A) ^\ n
for k being Element of NAT holds (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k
proof
let k be Element of NAT ; ::_thesis: (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k
dom (Prob * (A ^\ n)) = NAT by FUNCT_2:def_1;
then A26: (Prob * (A ^\ n)) . k = Prob . ((A ^\ n) . k) by FUNCT_1:12;
dom (Prob * A) = NAT by FUNCT_2:def_1;
then A27: Prob . (A . (n + k)) = (Prob * A) . (n + k) by FUNCT_1:12;
(Prob * A) . (k + n) = ((Prob * A) ^\ n) . k by NAT_1:def_3;
hence (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k by A26, A27, NAT_1:def_3; ::_thesis: verum
end;
hence Prob * (A ^\ n) = (Prob * A) ^\ n by FUNCT_2:63; ::_thesis: verum
end;
A28: for n being Element of NAT holds Partial_Sums (Prob * (A ^\ n)) is convergent
proof
let n be Element of NAT ; ::_thesis: Partial_Sums (Prob * (A ^\ n)) is convergent
Partial_Sums (Prob * (A ^\ n)) = Partial_Sums ((Prob * A) ^\ n) by A25;
hence Partial_Sums (Prob * (A ^\ n)) is convergent by A24; ::_thesis: verum
end;
A29: for n being Element of NAT holds lim (Prob * (Partial_Union (A ^\ n))) <= lim (Partial_Sums (Prob * (A ^\ n)))
proof
let n be Element of NAT ; ::_thesis: lim (Prob * (Partial_Union (A ^\ n))) <= lim (Partial_Sums (Prob * (A ^\ n)))
A30: for k being Element of NAT holds (Prob * (Partial_Union (A ^\ n))) . k <= (Partial_Sums (Prob * (A ^\ n))) . k by A8;
A31: Prob * (Partial_Union (A ^\ n)) is convergent by PROB_3:41;
Partial_Sums (Prob * (A ^\ n)) is convergent by A28;
hence lim (Prob * (Partial_Union (A ^\ n))) <= lim (Partial_Sums (Prob * (A ^\ n))) by A31, A30, SEQ_2:18; ::_thesis: verum
end;
A32: for n being Element of NAT holds Prob . (Union (A ^\ n)) <= lim (Partial_Sums (Prob * (A ^\ n)))
proof
let n be Element of NAT ; ::_thesis: Prob . (Union (A ^\ n)) <= lim (Partial_Sums (Prob * (A ^\ n)))
lim (Prob * (Partial_Union (A ^\ n))) <= lim (Partial_Sums (Prob * (A ^\ n))) by A29;
hence Prob . (Union (A ^\ n)) <= lim (Partial_Sums (Prob * (A ^\ n))) by PROB_3:41; ::_thesis: verum
end;
A33: for n being Element of NAT holds Prob . (Union (A ^\ n)) <= Sum (Prob * (A ^\ n))
proof
let n be Element of NAT ; ::_thesis: Prob . (Union (A ^\ n)) <= Sum (Prob * (A ^\ n))
lim (Partial_Sums (Prob * (A ^\ n))) = Sum (Prob * (A ^\ n)) by SERIES_1:def_3;
hence Prob . (Union (A ^\ n)) <= Sum (Prob * (A ^\ n)) by A32; ::_thesis: verum
end;
A34: for n being Element of NAT holds (Prob * (Union_Shift_Seq A)) . n <= (Sum_Shift_Seq (Prob,A)) . n
proof
let n be Element of NAT ; ::_thesis: (Prob * (Union_Shift_Seq A)) . n <= (Sum_Shift_Seq (Prob,A)) . n
A35: dom (Prob * (Union_Shift_Seq A)) = NAT by FUNCT_2:def_1;
A36: (Prob * (Union_Shift_Seq A)) . n = Prob . ((Union_Shift_Seq A) . n) by A35, FUNCT_1:12;
A37: Prob . (Union (A ^\ n)) <= Sum (Prob * (A ^\ n)) by A33;
Sum (Prob * (A ^\ n)) = (Sum_Shift_Seq (Prob,A)) . n by Def11;
hence (Prob * (Union_Shift_Seq A)) . n <= (Sum_Shift_Seq (Prob,A)) . n by Def7, A36, A37; ::_thesis: verum
end;
A38: 0 <= lim (Prob * (Partial_Intersection (Union_Shift_Seq A))) by A7, A3, SEQ_2:17;
A39: ( Sum_Shift_Seq (Prob,A) is convergent implies lim (Prob * (Partial_Intersection (Union_Shift_Seq A))) <= lim (Sum_Shift_Seq (Prob,A)) )
proof
assume A40: Sum_Shift_Seq (Prob,A) is convergent ; ::_thesis: lim (Prob * (Partial_Intersection (Union_Shift_Seq A))) <= lim (Sum_Shift_Seq (Prob,A))
A41: for n being Element of NAT holds (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n <= (Prob * (Union_Shift_Seq A)) . n
proof
let n be Element of NAT ; ::_thesis: (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n <= (Prob * (Union_Shift_Seq A)) . n
A42: Prob . ((Partial_Intersection (Union_Shift_Seq A)) . n) <= Prob . ((Union_Shift_Seq A) . n) by PROB_1:34, PROB_3:23;
A43: dom (Prob * (Partial_Intersection (Union_Shift_Seq A))) = NAT by FUNCT_2:def_1;
A44: dom (Prob * (Union_Shift_Seq A)) = NAT by FUNCT_2:def_1;
(Prob * (Union_Shift_Seq A)) . n = Prob . ((Union_Shift_Seq A) . n) by A44, FUNCT_1:12;
hence (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n <= (Prob * (Union_Shift_Seq A)) . n by A43, A42, FUNCT_1:12; ::_thesis: verum
end;
lim (Prob * (Partial_Intersection (Union_Shift_Seq A))) <= lim (Sum_Shift_Seq (Prob,A))
proof
A45: for n being Element of NAT holds (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n <= (Sum_Shift_Seq (Prob,A)) . n
proof
let n be Element of NAT ; ::_thesis: (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n <= (Sum_Shift_Seq (Prob,A)) . n
A46: (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n <= (Prob * (Union_Shift_Seq A)) . n by A41;
A47: (Prob * (Union_Shift_Seq A)) . n <= (Sum_Shift_Seq (Prob,A)) . n by A34;
thus (Prob * (Partial_Intersection (Union_Shift_Seq A))) . n <= (Sum_Shift_Seq (Prob,A)) . n by A46, A47, XXREAL_0:2; ::_thesis: verum
end;
thus lim (Prob * (Partial_Intersection (Union_Shift_Seq A))) <= lim (Sum_Shift_Seq (Prob,A)) by A7, A40, A45, SEQ_2:18; ::_thesis: verum
end;
hence lim (Prob * (Partial_Intersection (Union_Shift_Seq A))) <= lim (Sum_Shift_Seq (Prob,A)) ; ::_thesis: verum
end;
A48: for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds
( 0 = lim (Sum_Shift_Seq (Prob,A)) & Sum_Shift_Seq (Prob,A) is convergent )
proof
let A be SetSequence of Sigma; ::_thesis: ( Partial_Sums (Prob * A) is convergent implies ( 0 = lim (Sum_Shift_Seq (Prob,A)) & Sum_Shift_Seq (Prob,A) is convergent ) )
assume A49: Partial_Sums (Prob * A) is convergent ; ::_thesis: ( 0 = lim (Sum_Shift_Seq (Prob,A)) & Sum_Shift_Seq (Prob,A) is convergent )
then A50: Prob * A is summable by SERIES_1:def_2;
A51: for n being Element of NAT holds (Sum (Prob * A)) - (Sum ((Prob * A) ^\ (n + 1))) = (Partial_Sums (Prob * A)) . n
proof
let n be Element of NAT ; ::_thesis: (Sum (Prob * A)) - (Sum ((Prob * A) ^\ (n + 1))) = (Partial_Sums (Prob * A)) . n
(Sum (Prob * A)) - (Sum ((Prob * A) ^\ (n + 1))) = (((Partial_Sums (Prob * A)) . n) + (Sum ((Prob * A) ^\ (n + 1)))) - (Sum ((Prob * A) ^\ (n + 1))) by A50, SERIES_1:15;
hence (Sum (Prob * A)) - (Sum ((Prob * A) ^\ (n + 1))) = (Partial_Sums (Prob * A)) . n ; ::_thesis: verum
end;
A52: for n, m being Element of NAT st n <= m holds
abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n))
proof
let n, m be Element of NAT ; ::_thesis: ( n <= m implies abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) )
assume n <= m ; ::_thesis: abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n))
A53: ((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n) = ((Partial_Sums (Prob * A)) . m) - ((Sum (Prob * A)) - (Sum ((Prob * A) ^\ (n + 1)))) by A51;
A54: ((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n) = ((Sum (Prob * A)) - (Sum ((Prob * A) ^\ (m + 1)))) - ((Sum (Prob * A)) - (Sum ((Prob * A) ^\ (n + 1)))) by A51, A53;
A55: ((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n) = (Sum ((Prob * A) ^\ (n + 1))) - (Sum ((Prob * A) ^\ (m + 1))) by A54;
A56: for A being SetSequence of Sigma
for n being Element of NAT holds Prob * (A ^\ n) = (Prob * A) ^\ n
proof
let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds Prob * (A ^\ n) = (Prob * A) ^\ n
let n be Element of NAT ; ::_thesis: Prob * (A ^\ n) = (Prob * A) ^\ n
for k being Element of NAT holds (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k
proof
let k be Element of NAT ; ::_thesis: (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k
dom (Prob * (A ^\ n)) = NAT by FUNCT_2:def_1;
then A57: (Prob * (A ^\ n)) . k = Prob . ((A ^\ n) . k) by FUNCT_1:12;
dom (Prob * A) = NAT by FUNCT_2:def_1;
then A58: Prob . (A . (n + k)) = (Prob * A) . (n + k) by FUNCT_1:12;
(Prob * A) . (k + n) = ((Prob * A) ^\ n) . k by NAT_1:def_3;
hence (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k by A57, A58, NAT_1:def_3; ::_thesis: verum
end;
hence Prob * (A ^\ n) = (Prob * A) ^\ n by FUNCT_2:63; ::_thesis: verum
end;
A59: for n being Element of NAT holds ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = Sum ((Prob * A) ^\ (n + 1))
proof
let n be Element of NAT ; ::_thesis: ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = Sum ((Prob * A) ^\ (n + 1))
A60: ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = (Sum_Shift_Seq (Prob,A)) . (n + 1) by NAT_1:def_3;
(Sum_Shift_Seq (Prob,A)) . (n + 1) = Sum (Prob * (A ^\ (n + 1))) by Def11;
hence ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = Sum ((Prob * A) ^\ (n + 1)) by A56, A60; ::_thesis: verum
end;
A61: abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m))
proof
((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n) = (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (Sum ((Prob * A) ^\ (m + 1))) by A55, A59;
hence abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) by A59; ::_thesis: verum
end;
abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n))
proof
percases ( (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) = 0 or 0 < (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) or (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) < 0 ) ;
suppose (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) = 0 ; ::_thesis: abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n))
hence abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) ; ::_thesis: verum
end;
suppose 0 < (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) ; ::_thesis: abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n))
then A62: - 0 > - ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) ;
abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) = - ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) by A62, ABSVALUE:def_1;
hence abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) ; ::_thesis: verum
end;
supposeA63: (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) < 0 ; ::_thesis: abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n))
abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = - ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) by A63, ABSVALUE:def_1;
hence abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . m)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) ; ::_thesis: verum
end;
end;
end;
hence abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) by A61; ::_thesis: verum
end;
A64: ( ( for sr being real number st 0 < sr holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) < sr ) implies for sr being real number st 0 < sr holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr )
proof
assume A65: for sr being real number st 0 < sr holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) < sr ; ::_thesis: for sr being real number st 0 < sr holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr
let sr be real number ; ::_thesis: ( 0 < sr implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr )
assume A66: 0 < sr ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr
consider n being Element of NAT such that
A67: for m being Element of NAT st n <= m holds
abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) < sr by A65, A66;
take n ; ::_thesis: for m being Element of NAT st n <= m holds
abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr
let m be Element of NAT ; ::_thesis: ( n <= m implies abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr )
assume A68: n <= m ; ::_thesis: abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr
abs (((Partial_Sums (Prob * A)) . m) - ((Partial_Sums (Prob * A)) . n)) = abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) by A68, A52;
hence abs ((((Sum_Shift_Seq (Prob,A)) ^\ 1) . m) - (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n)) < sr by A67, A68; ::_thesis: verum
end;
A69: ( Partial_Sums (Prob * A) is convergent & (Sum_Shift_Seq (Prob,A)) ^\ 1 is convergent ) by A49, A64, SEQ_4:41;
A70: dom (((Sum_Shift_Seq (Prob,A)) ^\ 1) + (Partial_Sums (Prob * A))) = NAT by FUNCT_2:def_1;
consider B being Real_Sequence such that
A71: B = ((Sum_Shift_Seq (Prob,A)) ^\ 1) + (Partial_Sums (Prob * A)) ;
set B1 = NAT --> (Sum (Prob * A));
A72: Prob * A is summable by A49, SERIES_1:def_2;
A73: for n being Element of NAT holds (NAT --> (Sum (Prob * A))) . n = B . n
proof
let n be Element of NAT ; ::_thesis: (NAT --> (Sum (Prob * A))) . n = B . n
A74: for n being Element of NAT holds ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = Sum ((Prob * A) ^\ (n + 1))
proof
let n be Element of NAT ; ::_thesis: ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = Sum ((Prob * A) ^\ (n + 1))
A75: ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = (Sum_Shift_Seq (Prob,A)) . (n + 1) by NAT_1:def_3;
A76: for A being SetSequence of Sigma
for n being Element of NAT holds Prob * (A ^\ n) = (Prob * A) ^\ n
proof
let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds Prob * (A ^\ n) = (Prob * A) ^\ n
let n be Element of NAT ; ::_thesis: Prob * (A ^\ n) = (Prob * A) ^\ n
for k being Element of NAT holds (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k
proof
let k be Element of NAT ; ::_thesis: (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k
dom (Prob * (A ^\ n)) = NAT by FUNCT_2:def_1;
then A77: (Prob * (A ^\ n)) . k = Prob . ((A ^\ n) . k) by FUNCT_1:12;
dom (Prob * A) = NAT by FUNCT_2:def_1;
then A78: Prob . (A . (n + k)) = (Prob * A) . (n + k) by FUNCT_1:12;
(Prob * A) . (k + n) = ((Prob * A) ^\ n) . k by NAT_1:def_3;
hence (Prob * (A ^\ n)) . k = ((Prob * A) ^\ n) . k by A77, A78, NAT_1:def_3; ::_thesis: verum
end;
hence Prob * (A ^\ n) = (Prob * A) ^\ n by FUNCT_2:63; ::_thesis: verum
end;
(Sum_Shift_Seq (Prob,A)) . (n + 1) = Sum (Prob * (A ^\ (n + 1))) by Def11;
hence ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = Sum ((Prob * A) ^\ (n + 1)) by A75, A76; ::_thesis: verum
end;
A79: ((Sum_Shift_Seq (Prob,A)) ^\ 1) . n = Sum ((Prob * A) ^\ (n + 1)) by A74;
Sum (Prob * A) = ((Partial_Sums (Prob * A)) . n) + (Sum ((Prob * A) ^\ (n + 1))) by A72, SERIES_1:15;
then (NAT --> (Sum (Prob * A))) . n = ((Partial_Sums (Prob * A)) . n) + (((Sum_Shift_Seq (Prob,A)) ^\ 1) . n) by A79, FUNCOP_1:7;
hence (NAT --> (Sum (Prob * A))) . n = B . n by A70, A71, VALUED_1:def_1; ::_thesis: verum
end;
A80: lim (NAT --> (Sum (Prob * A))) = lim B
proof
ex k being Element of NAT st
for n being Element of NAT st k <= n holds
(NAT --> (Sum (Prob * A))) . n = B . n
proof
take 1 ; ::_thesis: for n being Element of NAT st 1 <= n holds
(NAT --> (Sum (Prob * A))) . n = B . n
thus for n being Element of NAT st 1 <= n holds
(NAT --> (Sum (Prob * A))) . n = B . n by A73; ::_thesis: verum
end;
hence lim (NAT --> (Sum (Prob * A))) = lim B by SEQ_4:19; ::_thesis: verum
end;
A81: Sum (Prob * A) = (NAT --> (Sum (Prob * A))) . 1 by FUNCOP_1:7
.= lim B by A80, SEQ_4:26 ;
A82: lim B = (lim ((Sum_Shift_Seq (Prob,A)) ^\ 1)) + (lim (Partial_Sums (Prob * A))) by A71, A69, SEQ_2:6;
Sum (Prob * A) = (lim ((Sum_Shift_Seq (Prob,A)) ^\ 1)) + (Sum (Prob * A)) by A81, A82, SERIES_1:def_3;
hence ( 0 = lim (Sum_Shift_Seq (Prob,A)) & Sum_Shift_Seq (Prob,A) is convergent ) by A69, SEQ_4:21, SEQ_4:22; ::_thesis: verum
end;
( lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) by A1, A48;
hence ( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) by A5, A7, A38, A39, PROB_2:def_1; ::_thesis: verum
end;
theorem Th14: :: BOR_CANT:14
for Omega being non empty set
for Sigma being SigmaField of Omega holds
( ( for X being set
for A being SetSequence of X
for n being Element of NAT
for x being set holds
( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st
( k >= n & x in A . k ) ) ) & ( for X being set
for A being SetSequence of X
for x being set holds
( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ) ) & ( for A being SetSequence of Sigma
for x being set holds
( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ) ) & ( for X being set
for A being SetSequence of X
for x being set holds
( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k ) ) & ( for A being SetSequence of Sigma
for x being set holds
( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k ) ) & ( for A being SetSequence of Sigma
for x being Element of Omega holds
( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
not x in A . k ) ) )
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega holds
( ( for X being set
for A being SetSequence of X
for n being Element of NAT
for x being set holds
( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st
( k >= n & x in A . k ) ) ) & ( for X being set
for A being SetSequence of X
for x being set holds
( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ) ) & ( for A being SetSequence of Sigma
for x being set holds
( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ) ) & ( for X being set
for A being SetSequence of X
for x being set holds
( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k ) ) & ( for A being SetSequence of Sigma
for x being set holds
( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k ) ) & ( for A being SetSequence of Sigma
for x being Element of Omega holds
( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
not x in A . k ) ) )
let Sigma be SigmaField of Omega; ::_thesis: ( ( for X being set
for A being SetSequence of X
for n being Element of NAT
for x being set holds
( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st
( k >= n & x in A . k ) ) ) & ( for X being set
for A being SetSequence of X
for x being set holds
( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ) ) & ( for A being SetSequence of Sigma
for x being set holds
( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ) ) & ( for X being set
for A being SetSequence of X
for x being set holds
( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k ) ) & ( for A being SetSequence of Sigma
for x being set holds
( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k ) ) & ( for A being SetSequence of Sigma
for x being Element of Omega holds
( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
not x in A . k ) ) )
A1: for X being set
for A being SetSequence of X
for n being Element of NAT
for x being set holds
( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st
( k >= n & x in A . k ) )
proof
let X be set ; ::_thesis: for A being SetSequence of X
for n being Element of NAT
for x being set holds
( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st
( k >= n & x in A . k ) )
let A be SetSequence of X; ::_thesis: for n being Element of NAT
for x being set holds
( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st
( k >= n & x in A . k ) )
let n be Element of NAT ; ::_thesis: for x being set holds
( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st
( k >= n & x in A . k ) )
let x be set ; ::_thesis: ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st
( k >= n & x in A . k ) )
hereby ::_thesis: ( ex k being Element of NAT st
( k >= n & x in A . k ) implies ex k being Element of NAT st x in (A ^\ n) . k )
assume ex k being Element of NAT st x in (A ^\ n) . k ; ::_thesis: ex k being Element of NAT st
( k >= n & x in A . k )
then consider k being Element of NAT such that
A2: x in (A ^\ n) . k ;
A3: x in A . (k + n) by A2, NAT_1:def_3;
consider k being Element of NAT such that
A4: x in A . (k + n) by A3;
consider k being Element of NAT such that
A5: ( k >= n & x in A . k ) by A4, NAT_1:11;
thus ex k being Element of NAT st
( k >= n & x in A . k ) by A5; ::_thesis: verum
end;
assume ex k being Element of NAT st
( k >= n & x in A . k ) ; ::_thesis: ex k being Element of NAT st x in (A ^\ n) . k
then consider k being Element of NAT such that
A6: ( k >= n & x in A . k ) ;
consider knat being Nat such that
A7: k = n + knat by A6, NAT_1:10;
reconsider knat = knat as Element of NAT by ORDINAL1:def_12;
( x in A . k implies x in (A ^\ n) . knat ) by A7, NAT_1:def_3;
hence ex k being Element of NAT st x in (A ^\ n) . k by A6; ::_thesis: verum
end;
A8: for X being set
for A being SetSequence of X
for x being set holds
( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) )
proof
let X be set ; ::_thesis: for A being SetSequence of X
for x being set holds
( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) )
let A be SetSequence of X; ::_thesis: for x being set holds
( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) )
let x be set ; ::_thesis: ( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) )
hereby ::_thesis: ( ( for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ) implies x in Intersection (Union_Shift_Seq A) )
assume A9: x in Intersection (Union_Shift_Seq A) ; ::_thesis: for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n )
A10: for n being Element of NAT st x in (Union_Shift_Seq A) . n holds
ex k being Element of NAT st
( k >= n & x in A . k )
proof
let n be Element of NAT ; ::_thesis: ( x in (Union_Shift_Seq A) . n implies ex k being Element of NAT st
( k >= n & x in A . k ) )
assume A11: x in (Union_Shift_Seq A) . n ; ::_thesis: ex k being Element of NAT st
( k >= n & x in A . k )
A12: ( x in (Union_Shift_Seq A) . n implies x in Union (A ^\ n) ) by Def7;
A13: ex k being Element of NAT st x in (A ^\ n) . k by A11, A12, PROB_1:12;
consider k being Element of NAT such that
A14: ( k >= n & x in A . k ) by A13, A1;
take k ; ::_thesis: ( k >= n & x in A . k )
thus ( k >= n & x in A . k ) by A14; ::_thesis: verum
end;
A15: for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n )
proof
let m be Element of NAT ; ::_thesis: ex n being Element of NAT st
( n >= m & x in A . n )
x in (Union_Shift_Seq A) . m by A9, PROB_1:13;
hence ex n being Element of NAT st
( n >= m & x in A . n ) by A10; ::_thesis: verum
end;
thus for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) by A15; ::_thesis: verum
end;
assume A16: for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ; ::_thesis: x in Intersection (Union_Shift_Seq A)
A17: for m being Element of NAT st ex n being Element of NAT st
( n >= m & x in A . n ) holds
x in (Union_Shift_Seq A) . m
proof
let m be Element of NAT ; ::_thesis: ( ex n being Element of NAT st
( n >= m & x in A . n ) implies x in (Union_Shift_Seq A) . m )
assume ex n being Element of NAT st
( n >= m & x in A . n ) ; ::_thesis: x in (Union_Shift_Seq A) . m
then consider n being Element of NAT such that
A18: ( n >= m & x in A . n ) ;
ex k being Element of NAT st x in (A ^\ m) . k by A18, A1;
then x in Union (A ^\ m) by PROB_1:12;
hence x in (Union_Shift_Seq A) . m by Def7; ::_thesis: verum
end;
for m being Element of NAT holds x in (Union_Shift_Seq A) . m
proof
let m be Element of NAT ; ::_thesis: x in (Union_Shift_Seq A) . m
ex n being Element of NAT st
( n >= m & x in A . n ) by A16;
hence x in (Union_Shift_Seq A) . m by A17; ::_thesis: verum
end;
hence x in Intersection (Union_Shift_Seq A) by PROB_1:13; ::_thesis: verum
end;
A19: for A being SetSequence of Sigma
for x being set holds
( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) )
proof
let A be SetSequence of Sigma; ::_thesis: for x being set holds
( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) )
let x be set ; ::_thesis: ( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) )
@Intersection (Union_Shift_Seq A) = Intersection (Union_Shift_Seq A) by PROB_2:def_1;
hence ( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ) by A8; ::_thesis: verum
end;
A20: for X being set
for A being SetSequence of X
for x being set holds
( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k )
proof
let X be set ; ::_thesis: for A being SetSequence of X
for x being set holds
( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k )
let A be SetSequence of X; ::_thesis: for x being set holds
( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k )
let x be set ; ::_thesis: ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k )
hereby ::_thesis: ( ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k implies x in Union (Intersect_Shift_Seq A) )
assume x in Union (Intersect_Shift_Seq A) ; ::_thesis: ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k
then consider n being Element of NAT such that
A21: x in (Intersect_Shift_Seq A) . n by PROB_1:12;
A22: (Intersect_Shift_Seq A) . n = Intersection (A ^\ n) by Def9;
for k being Element of NAT st k >= n holds
x in A . k
proof
let k be Element of NAT ; ::_thesis: ( k >= n implies x in A . k )
assume A23: n <= k ; ::_thesis: x in A . k
consider knat being Nat such that
A24: k = n + knat by A23, NAT_1:10;
reconsider knat = knat as Element of NAT by ORDINAL1:def_12;
A25: ( x in A . k iff x in (A ^\ n) . knat ) by A24, NAT_1:def_3;
thus x in A . k by A22, A21, A25, PROB_1:13; ::_thesis: verum
end;
hence ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k ; ::_thesis: verum
end;
assume ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k ; ::_thesis: x in Union (Intersect_Shift_Seq A)
then consider n being Element of NAT such that
A26: for k being Element of NAT st k >= n holds
x in A . k ;
set knat = the Nat;
for s being Element of NAT holds x in (A ^\ n) . s
proof
let s be Element of NAT ; ::_thesis: x in (A ^\ n) . s
( x in (A ^\ n) . s iff x in A . (n + s) ) by NAT_1:def_3;
hence x in (A ^\ n) . s by A26, NAT_1:12; ::_thesis: verum
end;
then x in Intersection (A ^\ n) by PROB_1:13;
then x in (Intersect_Shift_Seq A) . n by Def9;
hence x in Union (Intersect_Shift_Seq A) by PROB_1:12; ::_thesis: verum
end;
for A being SetSequence of Sigma
for x being Element of Omega holds
( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
not x in A . k )
proof
let A be SetSequence of Sigma; ::_thesis: for x being Element of Omega holds
( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
not x in A . k )
let x be Element of Omega; ::_thesis: ( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
not x in A . k )
hereby ::_thesis: ( ex n being Element of NAT st
for k being Element of NAT st k >= n holds
not x in A . k implies x in Union (Intersect_Shift_Seq (Complement A)) )
assume x in Union (Intersect_Shift_Seq (Complement A)) ; ::_thesis: ex n being Element of NAT st
for k being Element of NAT st k >= n holds
not x in A . k
then consider n being Element of NAT such that
A27: x in (Intersect_Shift_Seq (Complement A)) . n by PROB_1:12;
A28: (Intersect_Shift_Seq (Complement A)) . n = Intersection ((Complement A) ^\ n) by Def9;
set m = the Element of NAT ;
for k being Element of NAT st k >= n holds
not x in A . k
proof
let k be Element of NAT ; ::_thesis: ( k >= n implies not x in A . k )
assume A29: n <= k ; ::_thesis: not x in A . k
consider knat being Nat such that
A30: k = n + knat by A29, NAT_1:10;
reconsider knat = knat as Element of NAT by ORDINAL1:def_12;
A31: ( x in (Complement A) . k iff x in ((Complement A) ^\ n) . knat ) by A30, NAT_1:def_3;
x in (A . k) ` by A28, A27, A31, PROB_1:13, PROB_1:def_2;
then x in Omega \ (A . k) by SUBSET_1:def_4;
hence not x in A . k by XBOOLE_0:def_5; ::_thesis: verum
end;
hence ex n being Element of NAT st
for k being Element of NAT st k >= n holds
not x in A . k ; ::_thesis: verum
end;
assume ex n being Element of NAT st
for k being Element of NAT st k >= n holds
not x in A . k ; ::_thesis: x in Union (Intersect_Shift_Seq (Complement A))
then consider n being Element of NAT such that
A32: for k being Element of NAT st k >= n holds
not x in A . k ;
set k = the Element of NAT ;
A33: for k being Element of NAT st n <= k holds
x in (Complement A) . k
proof
let k be Element of NAT ; ::_thesis: ( n <= k implies x in (Complement A) . k )
assume A34: n <= k ; ::_thesis: x in (Complement A) . k
A35: not x in A . k by A34, A32;
x in Omega \ (A . k) by A35, XBOOLE_0:def_5;
then x in (A . k) ` by SUBSET_1:def_4;
hence x in (Complement A) . k by PROB_1:def_2; ::_thesis: verum
end;
for s being Element of NAT holds x in ((Complement A) ^\ n) . s
proof
let s be Element of NAT ; ::_thesis: x in ((Complement A) ^\ n) . s
( x in ((Complement A) ^\ n) . s iff x in (Complement A) . (n + s) ) by NAT_1:def_3;
hence x in ((Complement A) ^\ n) . s by A33, NAT_1:12; ::_thesis: verum
end;
then x in Intersection ((Complement A) ^\ n) by PROB_1:13;
then x in (Intersect_Shift_Seq (Complement A)) . n by Def9;
hence x in Union (Intersect_Shift_Seq (Complement A)) by PROB_1:12; ::_thesis: verum
end;
hence ( ( for X being set
for A being SetSequence of X
for n being Element of NAT
for x being set holds
( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st
( k >= n & x in A . k ) ) ) & ( for X being set
for A being SetSequence of X
for x being set holds
( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ) ) & ( for A being SetSequence of Sigma
for x being set holds
( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ) ) & ( for X being set
for A being SetSequence of X
for x being set holds
( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k ) ) & ( for A being SetSequence of Sigma
for x being set holds
( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k ) ) & ( for A being SetSequence of Sigma
for x being Element of Omega holds
( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st
for k being Element of NAT st k >= n holds
not x in A . k ) ) ) by A1, A8, A19, A20; ::_thesis: verum
end;
theorem Th15: :: BOR_CANT:15
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma holds
( lim_sup A = @lim_sup A & lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 )
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma holds
( lim_sup A = @lim_sup A & lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 )
let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma
for A being SetSequence of Sigma holds
( lim_sup A = @lim_sup A & lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 )
let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma holds
( lim_sup A = @lim_sup A & lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 )
let A be SetSequence of Sigma; ::_thesis: ( lim_sup A = @lim_sup A & lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 )
thus A1: lim_sup A = @lim_sup A ::_thesis: ( lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 )
proof
A2: for n being Element of NAT
for x being set holds
( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st
( k >= n & x in A . k ) )
proof
let n be Element of NAT ; ::_thesis: for x being set holds
( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st
( k >= n & x in A . k ) )
let x be set ; ::_thesis: ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st
( k >= n & x in A . k ) )
hereby ::_thesis: ( ex k being Element of NAT st
( k >= n & x in A . k ) implies ex k being Element of NAT st x in (A ^\ n) . k )
assume ex k being Element of NAT st x in (A ^\ n) . k ; ::_thesis: ex k being Element of NAT st
( k >= n & x in A . k )
then consider k being Element of NAT such that
A3: x in (A ^\ n) . k ;
A4: x in A . (k + n) by A3, NAT_1:def_3;
consider k being Element of NAT such that
A5: x in A . (k + n) by A4;
consider k being Element of NAT such that
A6: ( k >= n & x in A . k ) by A5, NAT_1:11;
thus ex k being Element of NAT st
( k >= n & x in A . k ) by A6; ::_thesis: verum
end;
assume ex k being Element of NAT st
( k >= n & x in A . k ) ; ::_thesis: ex k being Element of NAT st x in (A ^\ n) . k
then consider k being Element of NAT such that
A7: ( k >= n & x in A . k ) ;
consider knat being Nat such that
A8: k = n + knat by A7, NAT_1:10;
reconsider knat = knat as Element of NAT by ORDINAL1:def_12;
A9: ( x in A . k implies x in (A ^\ n) . knat ) by A8, NAT_1:def_3;
thus ex k being Element of NAT st x in (A ^\ n) . k by A7, A9; ::_thesis: verum
end;
A10: for x being set holds
( ( for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ) iff for m being Element of NAT ex n being Element of NAT st x in A . (m + n) )
proof
let x be set ; ::_thesis: ( ( for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ) iff for m being Element of NAT ex n being Element of NAT st x in A . (m + n) )
hereby ::_thesis: ( ( for m being Element of NAT ex n being Element of NAT st x in A . (m + n) ) implies for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) )
assume A11: for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ; ::_thesis: for m being Element of NAT ex n being Element of NAT st x in A . (m + n)
thus for m being Element of NAT ex n being Element of NAT st x in A . (m + n) ::_thesis: verum
proof
let m be Element of NAT ; ::_thesis: ex n being Element of NAT st x in A . (m + n)
ex n being Element of NAT st
( n >= m & x in A . n ) by A11;
then consider n being Element of NAT such that
A12: x in (A ^\ m) . n by A2;
x in A . (m + n) by A12, NAT_1:def_3;
hence ex n being Element of NAT st x in A . (m + n) ; ::_thesis: verum
end;
end;
assume A13: for m being Element of NAT ex n being Element of NAT st x in A . (m + n) ; ::_thesis: for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n )
thus for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) ::_thesis: verum
proof
let m be Element of NAT ; ::_thesis: ex n being Element of NAT st
( n >= m & x in A . n )
consider n being Element of NAT such that
A14: x in A . (m + n) by A13;
x in (A ^\ m) . n by A14, NAT_1:def_3;
hence ex n being Element of NAT st
( n >= m & x in A . n ) by A2; ::_thesis: verum
end;
end;
A15: for x being set holds
( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st x in A . (m + n) )
proof
let x be set ; ::_thesis: ( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st x in A . (m + n) )
hereby ::_thesis: ( ( for m being Element of NAT ex n being Element of NAT st x in A . (m + n) ) implies x in @Intersection (Union_Shift_Seq A) )
assume x in @Intersection (Union_Shift_Seq A) ; ::_thesis: for m being Element of NAT ex n being Element of NAT st x in A . (m + n)
then A16: for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) by Th14;
thus for m being Element of NAT ex n being Element of NAT st x in A . (m + n) by A16, A10; ::_thesis: verum
end;
assume for m being Element of NAT ex n being Element of NAT st x in A . (m + n) ; ::_thesis: x in @Intersection (Union_Shift_Seq A)
then for m being Element of NAT ex n being Element of NAT st
( n >= m & x in A . n ) by A10;
hence x in @Intersection (Union_Shift_Seq A) by Th14; ::_thesis: verum
end;
for x being set holds
( x in lim_sup A iff x in @Intersection (Union_Shift_Seq A) )
proof
let x be set ; ::_thesis: ( x in lim_sup A iff x in @Intersection (Union_Shift_Seq A) )
hereby ::_thesis: ( x in @Intersection (Union_Shift_Seq A) implies x in lim_sup A )
assume x in lim_sup A ; ::_thesis: x in @Intersection (Union_Shift_Seq A)
then A17: for m being Element of NAT ex n being Element of NAT st x in A . (m + n) by SETLIM_1:66;
thus x in @Intersection (Union_Shift_Seq A) by A17, A15; ::_thesis: verum
end;
assume x in @Intersection (Union_Shift_Seq A) ; ::_thesis: x in lim_sup A
then for m being Element of NAT ex n being Element of NAT st x in A . (m + n) by A15;
hence x in lim_sup A by SETLIM_1:66; ::_thesis: verum
end;
hence lim_sup A = @lim_sup A by TARSKI:1; ::_thesis: verum
end;
A18: for A being SetSequence of Sigma holds lim_inf A = @lim_inf A
proof
let A be SetSequence of Sigma; ::_thesis: lim_inf A = @lim_inf A
A19: for x being set holds
( ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k iff ex n being Element of NAT st
for k being Element of NAT holds x in A . (n + k) )
proof
let x be set ; ::_thesis: ( ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k iff ex n being Element of NAT st
for k being Element of NAT holds x in A . (n + k) )
hereby ::_thesis: ( ex n being Element of NAT st
for k being Element of NAT holds x in A . (n + k) implies ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k )
assume ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k ; ::_thesis: ex n being Element of NAT st
for k being Element of NAT holds x in A . (n + k)
then consider n being Element of NAT such that
A20: for k being Element of NAT st k >= n holds
x in A . k ;
for k being Element of NAT holds x in A . (n + k) by A20, NAT_1:11;
hence ex n being Element of NAT st
for k being Element of NAT holds x in A . (n + k) ; ::_thesis: verum
end;
assume ex n being Element of NAT st
for k being Element of NAT holds x in A . (n + k) ; ::_thesis: ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k
then consider n being Element of NAT such that
A21: for k being Element of NAT holds x in A . (n + k) ;
for k being Element of NAT st k >= n holds
x in A . k
proof
let k be Element of NAT ; ::_thesis: ( k >= n implies x in A . k )
assume n <= k ; ::_thesis: x in A . k
then consider knat being Nat such that
A22: k = n + knat by NAT_1:10;
reconsider knat = knat as Element of NAT by ORDINAL1:def_12;
x in A . (n + knat) by A21;
hence x in A . k by A22; ::_thesis: verum
end;
hence ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k ; ::_thesis: verum
end;
for x being set holds
( x in @lim_inf A iff x in lim_inf A )
proof
let x be set ; ::_thesis: ( x in @lim_inf A iff x in lim_inf A )
hereby ::_thesis: ( x in lim_inf A implies x in @lim_inf A )
assume x in @lim_inf A ; ::_thesis: x in lim_inf A
then ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k by Th14;
then ex n being Element of NAT st
for k being Element of NAT holds x in A . (n + k) by A19;
hence x in lim_inf A by SETLIM_1:67; ::_thesis: verum
end;
assume x in lim_inf A ; ::_thesis: x in @lim_inf A
then ex n being Element of NAT st
for k being Element of NAT holds x in A . (n + k) by SETLIM_1:67;
then ex n being Element of NAT st
for k being Element of NAT st k >= n holds
x in A . k by A19;
hence x in @lim_inf A by Th14; ::_thesis: verum
end;
hence lim_inf A = @lim_inf A by TARSKI:1; ::_thesis: verum
end;
A23: @lim_inf (Complement A) = (@lim_sup A) `
proof
reconsider CA = Complement A as SetSequence of Sigma ;
for x being set holds
( x in @lim_inf (Complement A) iff x in (@lim_sup A) ` )
proof
let x be set ; ::_thesis: ( x in @lim_inf (Complement A) iff x in (@lim_sup A) ` )
hereby ::_thesis: ( x in (@lim_sup A) ` implies x in @lim_inf (Complement A) )
assume x in @lim_inf (Complement A) ; ::_thesis: x in (@lim_sup A) `
then x in @lim_inf CA ;
then ( x in Omega & ex n being Element of NAT st
for k being Element of NAT st k >= n holds
not x in A . k ) by Th14;
then ( x in Omega & not x in @lim_sup A ) by Th14;
then x in Omega \ (@lim_sup A) by XBOOLE_0:def_5;
hence x in (@lim_sup A) ` by SUBSET_1:def_4; ::_thesis: verum
end;
assume A24: x in (@lim_sup A) ` ; ::_thesis: x in @lim_inf (Complement A)
x in Omega \ (@lim_sup A) by A24, SUBSET_1:def_4;
then not x in @Intersection (Union_Shift_Seq A) by XBOOLE_0:def_5;
then ex m being Element of NAT st
for n being Element of NAT st n >= m holds
not x in A . n by Th14;
then x in @lim_inf CA by A24, Th14;
hence x in @lim_inf (Complement A) ; ::_thesis: verum
end;
hence @lim_inf (Complement A) = (@lim_sup A) ` by TARSKI:1; ::_thesis: verum
end;
(Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1
proof
(Prob . (([#] Sigma) \ (@lim_sup A))) + (Prob . (@lim_sup A)) = 1 by PROB_1:31;
hence (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 by A23, SUBSET_1:def_4; ::_thesis: verum
end;
hence ( lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 ) by A1, A18, A23; ::_thesis: verum
end;
theorem Th16: :: BOR_CANT:16
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma holds
( ( Partial_Sums (Prob * A) is convergent implies ( Prob . (lim_sup A) = 0 & Prob . (lim_inf (Complement A)) = 1 ) ) & ( A is_all_independent_wrt Prob & Partial_Sums (Prob * A) is divergent_to+infty implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) )
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma holds
( ( Partial_Sums (Prob * A) is convergent implies ( Prob . (lim_sup A) = 0 & Prob . (lim_inf (Complement A)) = 1 ) ) & ( A is_all_independent_wrt Prob & Partial_Sums (Prob * A) is divergent_to+infty implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) )
let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma
for A being SetSequence of Sigma holds
( ( Partial_Sums (Prob * A) is convergent implies ( Prob . (lim_sup A) = 0 & Prob . (lim_inf (Complement A)) = 1 ) ) & ( A is_all_independent_wrt Prob & Partial_Sums (Prob * A) is divergent_to+infty implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) )
let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma holds
( ( Partial_Sums (Prob * A) is convergent implies ( Prob . (lim_sup A) = 0 & Prob . (lim_inf (Complement A)) = 1 ) ) & ( A is_all_independent_wrt Prob & Partial_Sums (Prob * A) is divergent_to+infty implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) )
let A be SetSequence of Sigma; ::_thesis: ( ( Partial_Sums (Prob * A) is convergent implies ( Prob . (lim_sup A) = 0 & Prob . (lim_inf (Complement A)) = 1 ) ) & ( A is_all_independent_wrt Prob & Partial_Sums (Prob * A) is divergent_to+infty implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) )
A1: ( Partial_Sums (Prob * A) is convergent implies Prob . (lim_inf (Complement A)) = 1 )
proof
assume A2: Partial_Sums (Prob * A) is convergent ; ::_thesis: Prob . (lim_inf (Complement A)) = 1
A3: Prob . (lim_inf (Complement A)) = Prob . (@lim_inf (Complement A)) by Th15;
for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds
( Prob . (@lim_inf (Complement A)) = 1 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent )
proof
let A be SetSequence of Sigma; ::_thesis: ( Partial_Sums (Prob * A) is convergent implies ( Prob . (@lim_inf (Complement A)) = 1 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) )
assume A4: Partial_Sums (Prob * A) is convergent ; ::_thesis: ( Prob . (@lim_inf (Complement A)) = 1 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent )
( (Prob . (@lim_sup A)) + (Prob . (@lim_inf (Complement A))) = 0 + (Prob . (@lim_inf (Complement A))) & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) by A4, Th13;
hence ( Prob . (@lim_inf (Complement A)) = 1 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent ) by Th15; ::_thesis: verum
end;
hence Prob . (lim_inf (Complement A)) = 1 by A2, A3; ::_thesis: verum
end;
A5: for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds
Prob . (lim_sup A) = 0
proof
let A be SetSequence of Sigma; ::_thesis: ( Partial_Sums (Prob * A) is convergent implies Prob . (lim_sup A) = 0 )
assume A6: Partial_Sums (Prob * A) is convergent ; ::_thesis: Prob . (lim_sup A) = 0
Prob . (lim_sup A) = Prob . (@lim_sup A) by Th15;
hence Prob . (lim_sup A) = 0 by A6, Th13; ::_thesis: verum
end;
for B being SetSequence of Sigma st B is_all_independent_wrt Prob & Partial_Sums (Prob * B) is divergent_to+infty holds
( Prob . (lim_inf (Complement B)) = 0 & Prob . (lim_sup B) = 1 )
proof
let B be SetSequence of Sigma; ::_thesis: ( B is_all_independent_wrt Prob & Partial_Sums (Prob * B) is divergent_to+infty implies ( Prob . (lim_inf (Complement B)) = 0 & Prob . (lim_sup B) = 1 ) )
assume that
A7: B is_all_independent_wrt Prob and
A8: Partial_Sums (Prob * B) is divergent_to+infty ; ::_thesis: ( Prob . (lim_inf (Complement B)) = 0 & Prob . (lim_sup B) = 1 )
A9: Prob . (@lim_sup B) = Prob . (lim_sup B) by Th15;
A10: Prob . (@lim_inf (Complement B)) = Prob . (lim_inf (Complement B)) by Th15;
for B being SetSequence of Sigma st B is_all_independent_wrt Prob & Partial_Sums (Prob * B) is divergent_to+infty holds
( Prob . (@lim_inf (Complement B)) = 0 & Prob . (@lim_sup B) = 1 )
proof
let B be SetSequence of Sigma; ::_thesis: ( B is_all_independent_wrt Prob & Partial_Sums (Prob * B) is divergent_to+infty implies ( Prob . (@lim_inf (Complement B)) = 0 & Prob . (@lim_sup B) = 1 ) )
assume that
A11: B is_all_independent_wrt Prob and
A12: Partial_Sums (Prob * B) is divergent_to+infty ; ::_thesis: ( Prob . (@lim_inf (Complement B)) = 0 & Prob . (@lim_sup B) = 1 )
A13: for Q being SetSequence of Sigma holds Intersect_Shift_Seq Q is non-descending
proof
let Q be SetSequence of Sigma; ::_thesis: Intersect_Shift_Seq Q is non-descending
inferior_setsequence Q = Intersect_Shift_Seq Q by Th11;
hence Intersect_Shift_Seq Q is non-descending ; ::_thesis: verum
end;
A14: Intersect_Shift_Seq (Complement B) is non-descending by A13;
reconsider CB = Complement B as SetSequence of Sigma ;
A15: Prob . (@lim_inf CB) = lim (Prob * (Intersect_Shift_Seq (Complement B))) by A14, PROB_2:10;
A16: for n being Element of NAT holds (Prob * (Intersect_Shift_Seq (Complement B))) . n = 0
proof
let n be Element of NAT ; ::_thesis: (Prob * (Intersect_Shift_Seq (Complement B))) . n = 0
dom (Prob * (Intersect_Shift_Seq (Complement B))) = NAT by FUNCT_2:def_1;
then A17: (Prob * (Intersect_Shift_Seq (Complement B))) . n = Prob . ((Intersect_Shift_Seq (Complement B)) . n) by FUNCT_1:12;
(Intersect_Shift_Seq (Complement B)) . n = Intersection ((Complement B) ^\ n) by Def9;
then A18: (Prob * (Intersect_Shift_Seq (Complement B))) . n = Prob . (Intersection (Partial_Intersection ((Complement B) ^\ n))) by A17, PROB_3:29;
Partial_Intersection ((Complement B) ^\ n) is non-ascending by PROB_3:27;
then A19: (Prob * (Intersect_Shift_Seq (Complement B))) . n = lim (Prob * (Partial_Intersection ((Complement B) ^\ n))) by A18, PROB_1:def_8;
A20: for k being Element of NAT holds (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k <= ((1 + (Partial_Sums (Prob * (B ^\ n)))) ") . k
proof
let k be Element of NAT ; ::_thesis: (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k <= ((1 + (Partial_Sums (Prob * (B ^\ n)))) ") . k
A21: for k being Element of NAT holds B ^\ k is_all_independent_wrt Prob
proof
let k be Element of NAT ; ::_thesis: B ^\ k is_all_independent_wrt Prob
for C being SetSequence of Sigma st ex e being sequence of NAT st
( e is one-to-one & ( for n being Element of NAT holds (B ^\ k) . (e . n) = C . n ) ) holds
for n being Element of NAT holds (Partial_Product (Prob * C)) . n = Prob . ((Partial_Intersection C) . n)
proof
let C be SetSequence of Sigma; ::_thesis: ( ex e being sequence of NAT st
( e is one-to-one & ( for n being Element of NAT holds (B ^\ k) . (e . n) = C . n ) ) implies for n being Element of NAT holds (Partial_Product (Prob * C)) . n = Prob . ((Partial_Intersection C) . n) )
given e being sequence of NAT such that A22: e is one-to-one and
A23: for n being Element of NAT holds (B ^\ k) . (e . n) = C . n ; ::_thesis: for n being Element of NAT holds (Partial_Product (Prob * C)) . n = Prob . ((Partial_Intersection C) . n)
A24: B ^\ k = B * (Special_Function2 k)
proof
for n being set st n in NAT holds
(B ^\ k) . n = (B * (Special_Function2 k)) . n
proof
let n be set ; ::_thesis: ( n in NAT implies (B ^\ k) . n = (B * (Special_Function2 k)) . n )
assume n in NAT ; ::_thesis: (B ^\ k) . n = (B * (Special_Function2 k)) . n
then reconsider n = n as Element of NAT ;
dom (B * (Special_Function2 k)) = NAT by FUNCT_2:def_1;
then A25: (B * (Special_Function2 k)) . n = B . ((Special_Function2 k) . n) by FUNCT_1:12;
(Special_Function2 k) . n = n + k by Def3;
hence (B ^\ k) . n = (B * (Special_Function2 k)) . n by A25, NAT_1:def_3; ::_thesis: verum
end;
hence B ^\ k = B * (Special_Function2 k) by FUNCT_2:12; ::_thesis: verum
end;
A26: for n being Element of NAT holds (B * (Special_Function2 k)) . (e . n) = B . (((Special_Function2 k) * e) . n)
proof
let n be Element of NAT ; ::_thesis: (B * (Special_Function2 k)) . (e . n) = B . (((Special_Function2 k) * e) . n)
( dom (B * (Special_Function2 k)) = NAT & dom ((Special_Function2 k) * e) = NAT ) by FUNCT_2:def_1;
then ( (B * (Special_Function2 k)) . (e . n) = B . ((Special_Function2 k) . (e . n)) & ((Special_Function2 k) * e) . n = (Special_Function2 k) . (e . n) ) by FUNCT_1:12;
hence (B * (Special_Function2 k)) . (e . n) = B . (((Special_Function2 k) * e) . n) ; ::_thesis: verum
end;
A27: for n being Element of NAT holds B . (((Special_Function2 k) * e) . n) = C . n
proof
let n be Element of NAT ; ::_thesis: B . (((Special_Function2 k) * e) . n) = C . n
(B * (Special_Function2 k)) . (e . n) = C . n by A24, A23;
hence B . (((Special_Function2 k) * e) . n) = C . n by A26; ::_thesis: verum
end;
(Special_Function2 k) * e is one-to-one by A22, FUNCT_1:24;
hence for n being Element of NAT holds (Partial_Product (Prob * C)) . n = Prob . ((Partial_Intersection C) . n) by A11, A27, Def6; ::_thesis: verum
end;
hence B ^\ k is_all_independent_wrt Prob by Def6; ::_thesis: verum
end;
A28: for A being SetSequence of Sigma
for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= ((1 + (Partial_Sums (Prob * A))) . n) "
proof
let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= ((1 + (Partial_Sums (Prob * A))) . n) "
let n be Element of NAT ; ::_thesis: (Partial_Product (Prob * (Complement A))) . n <= ((1 + (Partial_Sums (Prob * A))) . n) "
A29: (Partial_Product (Prob * (Complement A))) . n <= 1 / (1 + ((Partial_Sums (Prob * A)) . n))
proof
(Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n by Th4;
then A30: (Partial_Product (Prob * (Complement A))) . n <= exp_R . (- ((Partial_Sums (Prob * A)) . n)) by Th3;
exp_R . (- ((Partial_Sums (Prob * A)) . n)) <= 1 / (1 + ((Partial_Sums (Prob * A)) . n))
proof
A31: for n being Element of NAT holds (Prob * A) . n >= 0
proof
let n be Element of NAT ; ::_thesis: (Prob * A) . n >= 0
dom (Prob * A) = NAT by FUNCT_2:def_1;
then (Prob * A) . n = Prob . (A . n) by FUNCT_1:12;
hence (Prob * A) . n >= 0 by PROB_1:def_8; ::_thesis: verum
end;
A32: for n being Element of NAT holds (Partial_Sums (Prob * A)) . n >= 0
proof
let n be Element of NAT ; ::_thesis: (Partial_Sums (Prob * A)) . n >= 0
defpred S1[ Element of NAT ] means (Partial_Sums (Prob * A)) . $1 >= 0 ;
(Partial_Sums (Prob * A)) . 0 = (Prob * A) . 0 by SERIES_1:def_1;
then A33: S1[ 0 ] by A31;
A34: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A35: S1[k] ; ::_thesis: S1[k + 1]
A36: (Prob * A) . (k + 1) >= 0 by A31;
(Partial_Sums (Prob * A)) . (k + 1) = ((Partial_Sums (Prob * A)) . k) + ((Prob * A) . (k + 1)) by SERIES_1:def_1;
hence S1[k + 1] by A35, A36; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A33, A34);
hence (Partial_Sums (Prob * A)) . n >= 0 ; ::_thesis: verum
end;
for x being Element of REAL st x >= 0 holds
exp_R . (- x) <= 1 / (1 + x)
proof
let x be Element of REAL ; ::_thesis: ( x >= 0 implies exp_R . (- x) <= 1 / (1 + x) )
assume A37: x >= 0 ; ::_thesis: exp_R . (- x) <= 1 / (1 + x)
percases ( x > 0 or x <= 0 ) ;
supposeA38: x > 0 ; ::_thesis: exp_R . (- x) <= 1 / (1 + x)
A39: exp_R . (- x) >= 0 by SIN_COS:54;
set z = - x;
A40: (exp_R x) * (exp_R (- x)) = exp_R (x + (- x)) by SIN_COS:50;
(exp_R . (- x)) * (1 + x) <= 1 by Th2, A39, A40, SIN_COS:51, XREAL_1:64;
hence exp_R . (- x) <= 1 / (1 + x) by A38, XREAL_1:77; ::_thesis: verum
end;
suppose x <= 0 ; ::_thesis: exp_R . (- x) <= 1 / (1 + x)
then x = 0 by A37;
hence exp_R . (- x) <= 1 / (1 + x) by SIN_COS:51; ::_thesis: verum
end;
end;
end;
hence exp_R . (- ((Partial_Sums (Prob * A)) . n)) <= 1 / (1 + ((Partial_Sums (Prob * A)) . n)) by A32; ::_thesis: verum
end;
hence (Partial_Product (Prob * (Complement A))) . n <= 1 / (1 + ((Partial_Sums (Prob * A)) . n)) by A30, XXREAL_0:2; ::_thesis: verum
end;
for A being SetSequence of Sigma
for n being Element of NAT holds 1 / (1 + ((Partial_Sums (Prob * A)) . n)) = ((1 + (Partial_Sums (Prob * A))) . n) "
proof
let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds 1 / (1 + ((Partial_Sums (Prob * A)) . n)) = ((1 + (Partial_Sums (Prob * A))) . n) "
let n be Element of NAT ; ::_thesis: 1 / (1 + ((Partial_Sums (Prob * A)) . n)) = ((1 + (Partial_Sums (Prob * A))) . n) "
1 / (1 + ((Partial_Sums (Prob * A)) . n)) = 1 / ((1 + (Partial_Sums (Prob * A))) . n) by VALUED_1:2;
then 1 / (1 + ((Partial_Sums (Prob * A)) . n)) = 1 * (((1 + (Partial_Sums (Prob * A))) . n) ") by XCMPLX_0:def_9;
hence 1 / (1 + ((Partial_Sums (Prob * A)) . n)) = ((1 + (Partial_Sums (Prob * A))) . n) " ; ::_thesis: verum
end;
hence (Partial_Product (Prob * (Complement A))) . n <= ((1 + (Partial_Sums (Prob * A))) . n) " by A29; ::_thesis: verum
end;
dom (Prob * (Partial_Intersection (Complement (B ^\ n)))) = NAT by FUNCT_2:def_1;
then (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k = Prob . ((Partial_Intersection (Complement (B ^\ n))) . k) by FUNCT_1:12;
then (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k = (Partial_Product (Prob * (Complement (B ^\ n)))) . k by A21, Th10;
then (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k <= ((1 + (Partial_Sums (Prob * (B ^\ n)))) . k) " by A28;
hence (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k <= ((1 + (Partial_Sums (Prob * (B ^\ n)))) ") . k by VALUED_1:10; ::_thesis: verum
end;
A41: Partial_Sums (Prob * (B ^\ n)) is divergent_to+infty
proof
percases ( n = 0 or n <> 0 ) ;
suppose n = 0 ; ::_thesis: Partial_Sums (Prob * (B ^\ n)) is divergent_to+infty
hence Partial_Sums (Prob * (B ^\ n)) is divergent_to+infty by A12, NAT_1:47; ::_thesis: verum
end;
suppose n <> 0 ; ::_thesis: Partial_Sums (Prob * (B ^\ n)) is divergent_to+infty
then A42: n - 1 is Element of NAT by NAT_1:20;
consider y being Element of NAT such that
A43: y = n - 1 by A42;
set B2 = NAT --> (- ((Partial_Sums (Prob * B)) . y));
A44: (Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y))) is divergent_to+infty by A12, LIMFUNC1:18;
for r being Real ex q being Element of NAT st
for m being Element of NAT st q <= m holds
r < (Partial_Sums (Prob * (B ^\ n))) . m
proof
let r be Real; ::_thesis: ex q being Element of NAT st
for m being Element of NAT st q <= m holds
r < (Partial_Sums (Prob * (B ^\ n))) . m
for r being Real ex q being Element of NAT st
for m being Element of NAT st q <= m holds
r < (Partial_Sums (Prob * (B ^\ n))) . m
proof
let r be Real; ::_thesis: ex q being Element of NAT st
for m being Element of NAT st q <= m holds
r < (Partial_Sums (Prob * (B ^\ n))) . m
A45: for m being Element of NAT st n <= m holds
((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . m = (Partial_Sums (Prob * (B ^\ n))) . (m - n)
proof
let m be Element of NAT ; ::_thesis: ( n <= m implies ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . m = (Partial_Sums (Prob * (B ^\ n))) . (m - n) )
assume n <= m ; ::_thesis: ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . m = (Partial_Sums (Prob * (B ^\ n))) . (m - n)
then consider knat being Nat such that
A46: m = n + knat by NAT_1:10;
reconsider knat = knat as Element of NAT by ORDINAL1:def_12;
defpred S1[ Element of NAT ] means ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (n + $1) = (Partial_Sums (Prob * (B ^\ n))) . ((n + $1) - n);
A47: S1[ 0 ]
proof
dom ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) = NAT by FUNCT_2:def_1;
then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . n = ((Partial_Sums (Prob * B)) . n) + ((NAT --> (- ((Partial_Sums (Prob * B)) . y))) . n) by VALUED_1:def_1;
then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . n = ((Partial_Sums (Prob * B)) . n) + (- ((Partial_Sums (Prob * B)) . (n - 1))) by A43, FUNCOP_1:7;
then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . n = ((Partial_Sums (Prob * B)) . n) - ((Partial_Sums (Prob * B)) . (n - 1)) ;
then A48: ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . n = (((Partial_Sums (Prob * B)) . (n - 1)) + ((Prob * B) . ((n - 1) + 1))) - ((Partial_Sums (Prob * B)) . (n - 1)) by A42, SERIES_1:def_1;
dom (Prob * (B ^\ n)) = NAT by FUNCT_2:def_1;
then A49: (Prob * (B ^\ n)) . 0 = Prob . ((B ^\ n) . 0) by FUNCT_1:12;
A50: (B ^\ n) . 0 = B . (0 + n) by NAT_1:def_3;
dom (Prob * B) = NAT by FUNCT_2:def_1;
then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . n = (Prob * (B ^\ n)) . 0 by A50, A49, A48, FUNCT_1:12;
hence S1[ 0 ] by SERIES_1:def_1; ::_thesis: verum
end;
A51: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A52: S1[k] ; ::_thesis: S1[k + 1]
A53: dom ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) = NAT by FUNCT_2:def_1;
((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . ((n + k) + 1) = ((Partial_Sums (Prob * B)) . ((n + k) + 1)) + ((NAT --> (- ((Partial_Sums (Prob * B)) . y))) . ((n + k) + 1)) by A53, VALUED_1:def_1;
then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . ((n + k) + 1) = (((Partial_Sums (Prob * B)) . (n + k)) + ((Prob * B) . ((n + k) + 1))) + ((NAT --> (- ((Partial_Sums (Prob * B)) . y))) . ((n + k) + 1)) by SERIES_1:def_1;
then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . ((n + k) + 1) = (((Partial_Sums (Prob * B)) . (n + k)) + ((Prob * B) . ((n + k) + 1))) + (- ((Partial_Sums (Prob * B)) . (n - 1))) by A43, FUNCOP_1:7;
then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . ((n + k) + 1) = (((Partial_Sums (Prob * B)) . (n + k)) + ((Prob * B) . ((n + k) + 1))) + ((NAT --> (- ((Partial_Sums (Prob * B)) . y))) . (n + k)) by A43, FUNCOP_1:7;
then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . ((n + k) + 1) = (((Partial_Sums (Prob * B)) . (n + k)) + ((NAT --> (- ((Partial_Sums (Prob * B)) . y))) . (n + k))) + ((Prob * B) . ((n + k) + 1)) ;
then A54: ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . ((n + k) + 1) = ((Partial_Sums (Prob * (B ^\ n))) . ((n + k) - n)) + ((Prob * B) . ((n + k) + 1)) by A53, A52, VALUED_1:def_1;
(Prob * (B ^\ n)) . (((n + k) - n) + 1) = (Prob * B) . ((n + k) + 1)
proof
dom (Prob * (B ^\ n)) = NAT by FUNCT_2:def_1;
then A55: (Prob * (B ^\ n)) . (((n + k) - n) + 1) = Prob . ((B ^\ n) . (k + 1)) by FUNCT_1:12;
A56: (B ^\ n) . (k + 1) = B . (n + (k + 1)) by NAT_1:def_3;
dom (Prob * B) = NAT by FUNCT_2:def_1;
hence (Prob * (B ^\ n)) . (((n + k) - n) + 1) = (Prob * B) . ((n + k) + 1) by A56, A55, FUNCT_1:12; ::_thesis: verum
end;
hence S1[k + 1] by A54, SERIES_1:def_1; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A47, A51);
then ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (n + knat) = (Partial_Sums (Prob * (B ^\ n))) . ((n + knat) - n) ;
hence ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . m = (Partial_Sums (Prob * (B ^\ n))) . (m - n) by A46; ::_thesis: verum
end;
A57: ex q being Element of NAT st
for m being Element of NAT st q + n <= m + n holds
r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n)
proof
consider q being Element of NAT such that
A58: for m being Element of NAT st q <= m holds
r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . m by A44, LIMFUNC1:def_4;
for m being Element of NAT st q + n <= m + n holds
r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n)
proof
let m be Element of NAT ; ::_thesis: ( q + n <= m + n implies r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n) )
assume q + n <= m + n ; ::_thesis: r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n)
then ( q <= q + n & q + n <= m + n ) by NAT_1:11;
then q <= m + n by XXREAL_0:2;
hence r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n) by A58; ::_thesis: verum
end;
hence ex q being Element of NAT st
for m being Element of NAT st q + n <= m + n holds
r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n) ; ::_thesis: verum
end;
ex s being Element of NAT st
for m being Element of NAT st s <= m holds
r < (Partial_Sums (Prob * (B ^\ n))) . m
proof
consider q being Element of NAT such that
A59: for m being Element of NAT st q + n <= m + n holds
r < ((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n) by A57;
take s = q + n; ::_thesis: for m being Element of NAT st s <= m holds
r < (Partial_Sums (Prob * (B ^\ n))) . m
let m be Element of NAT ; ::_thesis: ( s <= m implies r < (Partial_Sums (Prob * (B ^\ n))) . m )
assume A60: s <= m ; ::_thesis: r < (Partial_Sums (Prob * (B ^\ n))) . m
set z = m + n;
((Partial_Sums (Prob * B)) + (NAT --> (- ((Partial_Sums (Prob * B)) . y)))) . (m + n) = (Partial_Sums (Prob * (B ^\ n))) . ((m + n) - n) by A45, NAT_1:12;
hence r < (Partial_Sums (Prob * (B ^\ n))) . m by A60, A59, NAT_1:12; ::_thesis: verum
end;
hence ex q being Element of NAT st
for m being Element of NAT st q <= m holds
r < (Partial_Sums (Prob * (B ^\ n))) . m ; ::_thesis: verum
end;
hence ex q being Element of NAT st
for m being Element of NAT st q <= m holds
r < (Partial_Sums (Prob * (B ^\ n))) . m ; ::_thesis: verum
end;
hence Partial_Sums (Prob * (B ^\ n)) is divergent_to+infty by LIMFUNC1:def_4; ::_thesis: verum
end;
end;
end;
A61: for A being SetSequence of Sigma st Partial_Sums (Prob * A) is divergent_to+infty holds
( lim ((1 + (Partial_Sums (Prob * A))) ") = 0 & (1 + (Partial_Sums (Prob * A))) " is convergent )
proof
let A be SetSequence of Sigma; ::_thesis: ( Partial_Sums (Prob * A) is divergent_to+infty implies ( lim ((1 + (Partial_Sums (Prob * A))) ") = 0 & (1 + (Partial_Sums (Prob * A))) " is convergent ) )
A62: for A being SetSequence of Sigma st ( for r being Real ex n being Element of NAT st
for m being Element of NAT st n <= m holds
r < (Partial_Sums (Prob * A)) . m ) holds
for r being Real ex n being Element of NAT st
for m being Element of NAT st n <= m holds
r < (1 + (Partial_Sums (Prob * A))) . m
proof
let A be SetSequence of Sigma; ::_thesis: ( ( for r being Real ex n being Element of NAT st
for m being Element of NAT st n <= m holds
r < (Partial_Sums (Prob * A)) . m ) implies for r being Real ex n being Element of NAT st
for m being Element of NAT st n <= m holds
r < (1 + (Partial_Sums (Prob * A))) . m )
assume A63: for r being Real ex n being Element of NAT st
for m being Element of NAT st n <= m holds
r < (Partial_Sums (Prob * A)) . m ; ::_thesis: for r being Real ex n being Element of NAT st
for m being Element of NAT st n <= m holds
r < (1 + (Partial_Sums (Prob * A))) . m
let r be Real; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
r < (1 + (Partial_Sums (Prob * A))) . m
consider n being Element of NAT such that
A64: for m being Element of NAT st n <= m holds
r < (Partial_Sums (Prob * A)) . m by A63;
take n ; ::_thesis: for m being Element of NAT st n <= m holds
r < (1 + (Partial_Sums (Prob * A))) . m
for m being Element of NAT st n <= m holds
r < (1 + (Partial_Sums (Prob * A))) . m
proof
let m be Element of NAT ; ::_thesis: ( n <= m implies r < (1 + (Partial_Sums (Prob * A))) . m )
assume n <= m ; ::_thesis: r < (1 + (Partial_Sums (Prob * A))) . m
then A65: r < (Partial_Sums (Prob * A)) . m by A64;
A66: (Partial_Sums (Prob * A)) . m < ((Partial_Sums (Prob * A)) . m) + 1 by XREAL_1:29;
(1 + (Partial_Sums (Prob * A))) . m = ((Partial_Sums (Prob * A)) . m) + 1 by VALUED_1:2;
hence r < (1 + (Partial_Sums (Prob * A))) . m by A65, A66, XXREAL_0:2; ::_thesis: verum
end;
hence for m being Element of NAT st n <= m holds
r < (1 + (Partial_Sums (Prob * A))) . m ; ::_thesis: verum
end;
assume Partial_Sums (Prob * A) is divergent_to+infty ; ::_thesis: ( lim ((1 + (Partial_Sums (Prob * A))) ") = 0 & (1 + (Partial_Sums (Prob * A))) " is convergent )
then for r being Real ex n being Element of NAT st
for m being Element of NAT st n <= m holds
r < (Partial_Sums (Prob * A)) . m by LIMFUNC1:def_4;
then for r being Real ex n being Element of NAT st
for m being Element of NAT st n <= m holds
r < (1 + (Partial_Sums (Prob * A))) . m by A62;
then 1 + (Partial_Sums (Prob * A)) is divergent_to+infty by LIMFUNC1:def_4;
hence ( lim ((1 + (Partial_Sums (Prob * A))) ") = 0 & (1 + (Partial_Sums (Prob * A))) " is convergent ) by LIMFUNC1:34; ::_thesis: verum
end;
Partial_Intersection (Complement (B ^\ n)) is non-ascending by PROB_3:27;
then A67: ( Prob * (Partial_Intersection (Complement (B ^\ n))) is convergent & (1 + (Partial_Sums (Prob * (B ^\ n)))) " is convergent ) by A41, A61, PROB_1:def_8;
A68: lim ((1 + (Partial_Sums (Prob * (B ^\ n)))) ") = 0 by A41, A61;
A69: for k being Element of NAT holds 0 <= (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k
proof
let k be Element of NAT ; ::_thesis: 0 <= (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k
dom (Prob * (Partial_Intersection (Complement (B ^\ n)))) = NAT by FUNCT_2:def_1;
then (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k = Prob . ((Partial_Intersection (Complement (B ^\ n))) . k) by FUNCT_1:12;
hence 0 <= (Prob * (Partial_Intersection (Complement (B ^\ n)))) . k by PROB_1:def_8; ::_thesis: verum
end;
A70: lim (Prob * (Partial_Intersection (Complement (B ^\ n)))) <= 0 by A67, A20, A68, SEQ_2:18;
Complement (B ^\ n) = (Complement B) ^\ n
proof
for k being set st k in NAT holds
(Complement (B ^\ n)) . k = ((Complement B) ^\ n) . k
proof
let k be set ; ::_thesis: ( k in NAT implies (Complement (B ^\ n)) . k = ((Complement B) ^\ n) . k )
assume k in NAT ; ::_thesis: (Complement (B ^\ n)) . k = ((Complement B) ^\ n) . k
then reconsider k = k as Element of NAT ;
A71: (Complement (B ^\ n)) . k = ((B ^\ n) . k) ` by PROB_1:def_2;
((Complement B) ^\ n) . k = (Complement B) . (k + n) by NAT_1:def_3;
then ((Complement B) ^\ n) . k = (B . (k + n)) ` by PROB_1:def_2;
hence (Complement (B ^\ n)) . k = ((Complement B) ^\ n) . k by A71, NAT_1:def_3; ::_thesis: verum
end;
hence Complement (B ^\ n) = (Complement B) ^\ n by FUNCT_2:12; ::_thesis: verum
end;
hence (Prob * (Intersect_Shift_Seq (Complement B))) . n = 0 by A69, A67, A70, A19, SEQ_2:17; ::_thesis: verum
end;
set B2 = NAT --> 0;
A72: ex n being Element of NAT st (NAT --> 0) . n = 0
proof
take 1 ; ::_thesis: (NAT --> 0) . 1 = 0
thus (NAT --> 0) . 1 = 0 by FUNCOP_1:7; ::_thesis: verum
end;
A73: lim (NAT --> 0) = 0 by A72, SEQ_4:25;
A74: ( NAT --> 0 is convergent & ex k being Element of NAT st
for n being Element of NAT st k <= n holds
(NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n )
proof
ex k being Element of NAT st
for n being Element of NAT st k <= n holds
(NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n
proof
A75: for n being Element of NAT st n >= 0 holds
(NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n
proof
let n be Element of NAT ; ::_thesis: ( n >= 0 implies (NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n )
assume n >= 0 ; ::_thesis: (NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n
( (NAT --> 0) . n = 0 & (Prob * (Intersect_Shift_Seq (Complement B))) . n = 0 ) by A16, FUNCOP_1:7;
hence (NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n ; ::_thesis: verum
end;
take 0 ; ::_thesis: for n being Element of NAT st 0 <= n holds
(NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n
thus for n being Element of NAT st 0 <= n holds
(NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n by A75; ::_thesis: verum
end;
hence ( NAT --> 0 is convergent & ex k being Element of NAT st
for n being Element of NAT st k <= n holds
(NAT --> 0) . n = (Prob * (Intersect_Shift_Seq (Complement B))) . n ) ; ::_thesis: verum
end;
( Prob . (@lim_inf (Complement B)) = 0 & (Prob . (@lim_inf (Complement B))) + (Prob . (@lim_sup B)) = 1 ) by A15, A74, A73, Th15, SEQ_4:19;
hence ( Prob . (@lim_inf (Complement B)) = 0 & Prob . (@lim_sup B) = 1 ) ; ::_thesis: verum
end;
hence ( Prob . (lim_inf (Complement B)) = 0 & Prob . (lim_sup B) = 1 ) by A9, A10, A7, A8; ::_thesis: verum
end;
hence ( ( Partial_Sums (Prob * A) is convergent implies ( Prob . (lim_sup A) = 0 & Prob . (lim_inf (Complement A)) = 1 ) ) & ( A is_all_independent_wrt Prob & Partial_Sums (Prob * A) is divergent_to+infty implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) ) by A5, A1; ::_thesis: verum
end;
theorem Th17: :: BOR_CANT:17
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma st not Partial_Sums (Prob * A) is convergent & A is_all_independent_wrt Prob holds
( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 )
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma st not Partial_Sums (Prob * A) is convergent & A is_all_independent_wrt Prob holds
( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 )
let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma
for A being SetSequence of Sigma st not Partial_Sums (Prob * A) is convergent & A is_all_independent_wrt Prob holds
( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 )
let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma st not Partial_Sums (Prob * A) is convergent & A is_all_independent_wrt Prob holds
( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 )
let A be SetSequence of Sigma; ::_thesis: ( not Partial_Sums (Prob * A) is convergent & A is_all_independent_wrt Prob implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) )
assume A1: not Partial_Sums (Prob * A) is convergent ; ::_thesis: ( not A is_all_independent_wrt Prob or ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) )
assume A2: A is_all_independent_wrt Prob ; ::_thesis: ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 )
A3: for n being Element of NAT holds (Prob * A) . n >= 0
proof
let n be Element of NAT ; ::_thesis: (Prob * A) . n >= 0
dom (Prob * A) = NAT by FUNCT_2:def_1;
then (Prob * A) . n = Prob . (A . n) by FUNCT_1:12;
hence (Prob * A) . n >= 0 by PROB_1:def_8; ::_thesis: verum
end;
A4: ( ( not Prob * A is summable implies not Partial_Sums (Prob * A) is bounded_above ) & not Prob * A is summable ) by A3, A1, SERIES_1:17, SERIES_1:def_2;
Partial_Sums (Prob * A) is divergent_to+infty by A4, A3, LIMFUNC1:29, SERIES_1:16;
hence ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) by A2, Th16; ::_thesis: verum
end;
theorem :: BOR_CANT:18
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma st A is_all_independent_wrt Prob holds
( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) )
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma st A is_all_independent_wrt Prob holds
( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) )
let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma
for A being SetSequence of Sigma st A is_all_independent_wrt Prob holds
( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) )
let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma st A is_all_independent_wrt Prob holds
( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) )
let A be SetSequence of Sigma; ::_thesis: ( A is_all_independent_wrt Prob implies ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) ) )
assume A1: A is_all_independent_wrt Prob ; ::_thesis: ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) )
percases ( Partial_Sums (Prob * A) is convergent or not Partial_Sums (Prob * A) is convergent ) ;
suppose Partial_Sums (Prob * A) is convergent ; ::_thesis: ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) )
hence ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) ) by Th16; ::_thesis: verum
end;
suppose not Partial_Sums (Prob * A) is convergent ; ::_thesis: ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) )
hence ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) ) by A1, Th17; ::_thesis: verum
end;
end;
end;
theorem :: BOR_CANT:19
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n1, n being Element of NAT holds (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1)
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n1, n being Element of NAT holds (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1)
let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n1, n being Element of NAT holds (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1)
let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma
for n1, n being Element of NAT holds (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1)
let A be SetSequence of Sigma; ::_thesis: for n1, n being Element of NAT holds (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1)
let n1, n be Element of NAT ; ::_thesis: (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1)
A1: dom (Prob * (A ^\ (n1 + 1))) = NAT by FUNCT_2:def_1;
A2: dom (Prob * A) = NAT by FUNCT_2:def_1;
defpred S1[ Element of NAT ] means (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . $1 <= ((Partial_Sums (Prob * A)) . (($1 + n1) + 1)) - ((Partial_Sums (Prob * A)) . n1);
A3: ((Partial_Sums (Prob * A)) . (n1 + 1)) - ((Partial_Sums (Prob * A)) . n1) = (((Partial_Sums (Prob * A)) . n1) + ((Prob * A) . (n1 + 1))) - ((Partial_Sums (Prob * A)) . n1) by SERIES_1:def_1;
A4: Prob . ((A ^\ (n1 + 1)) . 0) = Prob . (A . ((n1 + 1) + 0)) by NAT_1:def_3;
A5: Prob . (A . (n1 + 1)) = (Prob * A) . (n1 + 1) by A2, FUNCT_1:12;
A6: (Prob * (A ^\ (n1 + 1))) . 0 = (Prob * A) . (n1 + 1) by A1, A4, A5, FUNCT_1:12;
A7: S1[ 0 ] by A6, A3, SERIES_1:def_1;
A8: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A9: (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . k <= ((Partial_Sums (Prob * A)) . ((k + n1) + 1)) - ((Partial_Sums (Prob * A)) . n1) ; ::_thesis: S1[k + 1]
A10: ((Partial_Sums (Prob * (A ^\ (n1 + 1)))) . k) + ((Prob * (A ^\ (n1 + 1))) . (k + 1)) <= (((Partial_Sums (Prob * A)) . ((k + n1) + 1)) - ((Partial_Sums (Prob * A)) . n1)) + ((Prob * (A ^\ (n1 + 1))) . (k + 1)) by A9, XREAL_1:6;
A11: (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . (k + 1) <= (((Partial_Sums (Prob * A)) . ((k + n1) + 1)) - ((Partial_Sums (Prob * A)) . n1)) + ((Prob * (A ^\ (n1 + 1))) . (k + 1)) by A10, SERIES_1:def_1;
A12: ((Partial_Sums (Prob * (A ^\ (n1 + 1)))) . (k + 1)) + ((Partial_Sums (Prob * A)) . n1) <= ((((Partial_Sums (Prob * A)) . ((k + n1) + 1)) + ((Prob * (A ^\ (n1 + 1))) . (k + 1))) - ((Partial_Sums (Prob * A)) . n1)) + ((Partial_Sums (Prob * A)) . n1) by A11, XREAL_1:6;
A13: (((Partial_Sums (Prob * (A ^\ (n1 + 1)))) . (k + 1)) + ((Partial_Sums (Prob * A)) . n1)) - ((Partial_Sums (Prob * A)) . ((k + n1) + 1)) <= (((Prob * (A ^\ (n1 + 1))) . (k + 1)) + ((Partial_Sums (Prob * A)) . ((k + n1) + 1))) - ((Partial_Sums (Prob * A)) . ((k + n1) + 1)) by A12, XREAL_1:9;
A14: (A ^\ (n1 + 1)) . (k + 1) = A . ((n1 + 1) + (k + 1)) by NAT_1:def_3;
A15: dom (Prob * A) = NAT by FUNCT_2:def_1;
A16: dom (Prob * (A ^\ (n1 + 1))) = NAT by FUNCT_2:def_1;
A17: Prob . ((A ^\ (n1 + 1)) . (k + 1)) = (Prob * (A ^\ (n1 + 1))) . (k + 1) by A16, FUNCT_1:12;
A18: (((Partial_Sums (Prob * (A ^\ (n1 + 1)))) . (k + 1)) + ((Partial_Sums (Prob * A)) . n1)) - ((Partial_Sums (Prob * A)) . ((k + n1) + 1)) <= (Prob * A) . ((n1 + k) + 2) by A13, A17, A14, A15, FUNCT_1:12;
A19: ((((Partial_Sums (Prob * (A ^\ (n1 + 1)))) . (k + 1)) + ((Partial_Sums (Prob * A)) . n1)) - ((Partial_Sums (Prob * A)) . ((k + n1) + 1))) + ((Partial_Sums (Prob * A)) . ((k + n1) + 1)) <= ((Prob * A) . ((n1 + k) + 2)) + ((Partial_Sums (Prob * A)) . ((k + n1) + 1)) by A18, XREAL_1:6;
A20: (Partial_Sums (Prob * A)) . (((k + n1) + 1) + 1) = ((Partial_Sums (Prob * A)) . ((k + n1) + 1)) + ((Prob * A) . (((k + n1) + 1) + 1)) by SERIES_1:def_1;
(((Partial_Sums (Prob * (A ^\ (n1 + 1)))) . (k + 1)) + ((Partial_Sums (Prob * A)) . n1)) - ((Partial_Sums (Prob * A)) . n1) <= ((Partial_Sums (Prob * A)) . ((k + n1) + 2)) - ((Partial_Sums (Prob * A)) . n1) by A19, A20, XREAL_1:9;
hence S1[k + 1] ; ::_thesis: verum
end;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A7, A8);
then S1[n] ;
hence (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1) ; ::_thesis: verum
end;
theorem :: BOR_CANT:20
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n))
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n))
let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n))
let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma
for n being Element of NAT holds Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n))
let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n))
let n be Element of NAT ; ::_thesis: Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n))
A1: Prob . ((Intersect_Shift_Seq (Complement A)) . n) = Prob . (((Union_Shift_Seq A) . n) `) by Th9;
Prob . (((Union_Shift_Seq A) . n) `) = Prob . (([#] Sigma) \ ((Union_Shift_Seq A) . n)) by SUBSET_1:def_4;
hence Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n)) by A1, PROB_1:32; ::_thesis: verum
end;
theorem :: BOR_CANT:21
for Omega being non empty set
for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds
( ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) & ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ) )
proof
let Omega be non empty set ; ::_thesis: for Sigma being SigmaField of Omega
for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds
( ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) & ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ) )
let Sigma be SigmaField of Omega; ::_thesis: for Prob being Probability of Sigma
for A being SetSequence of Sigma
for n being Element of NAT holds
( ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) & ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ) )
let Prob be Probability of Sigma; ::_thesis: for A being SetSequence of Sigma
for n being Element of NAT holds
( ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) & ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ) )
let A be SetSequence of Sigma; ::_thesis: for n being Element of NAT holds
( ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) & ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ) )
let n be Element of NAT ; ::_thesis: ( ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) & ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ) )
thus ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) ::_thesis: ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n )
proof
assume A1: Complement A is_all_independent_wrt Prob ; ::_thesis: Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n
( (Partial_Intersection (Complement (Complement A))) . n = (Partial_Intersection A) . n & (Partial_Product (Prob * (Complement (Complement A)))) . n = (Partial_Product (Prob * A)) . n & Prob . ((Partial_Intersection (Complement (Complement A))) . n) = (Partial_Product (Prob * (Complement (Complement A)))) . n ) by A1, Th10;
hence Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ; ::_thesis: verum
end;
assume A is_all_independent_wrt Prob ; ::_thesis: 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n
then ( Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n & Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n)) ) by Th10, Th8;
hence 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ; ::_thesis: verum
end;