:: Events of Borel Sets, Construction of Borel Sets and Random Variables
:: for Stochastic Finance
:: by Peter Jaeger
::
:: Received July 10, 2014
:: Copyright (c) 2014-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies FINANCE1, PROB_3, SERIES_1, SEQ_1, XXREAL_1, RANDOM_3, FINANCE2,
NUMBERS, XBOOLE_0, PROB_1, SUBSET_1, REAL_1, TARSKI, RELAT_1, FUNCT_1,
ARYTM_3, XXREAL_0, BORSUK_5, VALUED_1, ARYTM_1, NAT_1, CARD_3, CARD_1,
ZFMISC_1, RPR_1, POWER, RANDOM_1, FUNCT_7, SETFAM_1, INT_1, RAT_1,
LIMFUNC1;
notations TARSKI, SUBSET_1, XBOOLE_0,
ORDINAL1, PROB_3, SERIES_1, SEQ_1, XXREAL_1,
RCOMP_1, SETFAM_1, RELAT_1, FINANCE1, RANDOM_3, NUMBERS, XREAL_0,
XXREAL_0, XCMPLX_0, FUNCT_1, REAL_1, VALUED_1, NAT_1, FUNCT_2, PROB_1,
LIMFUNC1, RANDOM_1, FINSUB_1, INT_1, RAT_1, IRRAT_1, SEQ_4, BORSUK_5;
constructors PROB_3, SERIES_1, FINANCE1, RANDOM_3, REAL_1, RELSET_1, RCOMP_1,
SEQ_4, BORSUK_5, LIMFUNC1;
registrations PROB_1, FINANCE1, SUBSET_1, NAT_1, XREAL_0, XXREAL_0, MEMBERED,
ORDINAL1, FUNCT_2, NUMBERS, VALUED_0, VALUED_1, RELSET_1, FINSUB_1,
INT_1, BORSUK_5, RAT_1;
requirements NUMERALS, REAL, BOOLE, SUBSET, ARITHM;
definitions TARSKI;
equalities PROB_1, FINANCE1, SUBSET_1, BORSUK_5, XCMPLX_0, LIMFUNC1;
expansions FUNCT_2, MEMBERED, TARSKI, XBOOLE_0, PROB_1;
theorems SERIES_1, PROB_3, XXREAL_1, FINANCE1, RANDOM_3, TARSKI, XBOOLE_0,
XBOOLE_1, XCMPLX_1, XXREAL_0, PROB_1, NAT_1, FUNCT_2, ORDINAL1, VALUED_1,
XREAL_0, FUNCT_1, NUMBERS, SUBSET_1, INT_1, ZFMISC_1, XCMPLX_0, RAT_1;
schemes NAT_1, RECDEF_1, FUNCT_2;
begin :: Events of Borel-Sets
:: We start with some theorems which are necessary to prove, that IRRAT is an
:: Event of the Borel-Sets
theorem Ko1:
1 in REAL & -1 in REAL
proof
reconsider a = 1 as Element of REAL by NUMBERS:19;
consider myp1 being Element of REAL such that A2: myp1 = -a;
thus thesis by A2;
end;
theorem Th5:
number_e in REAL\RAT
proof
number_e in REAL & not number_e in RAT by XREAL_0:def 1;
hence thesis by XBOOLE_0:def 5;
end;
theorem Th6:
number_e in REAL\INT
proof
number_e in REAL & not number_e in RAT by XREAL_0:def 1;
then number_e in REAL\RAT by XBOOLE_0:def 5;
hence thesis by NUMBERS:14,XBOOLE_1:34,TARSKI:def 3;
end;
theorem Th7:
number_e in REAL\NAT
proof
number_e in REAL & not number_e in RAT by XREAL_0:def 1;
then number_e in REAL\RAT by XBOOLE_0:def 5;
hence thesis by NUMBERS:18,XBOOLE_1:34,TARSKI:def 3;
end;
registration
cluster REAL \ RAT -> non empty;
coherence by Th5;
cluster REAL \ INT -> non empty;
coherence by Th6;
cluster REAL \ NAT -> non empty;
coherence by Th7;
end;
theorem Th1:
for k being Real holds {k} in Borel_Sets
proof
let k be Real;
[.k,k.] is Element of Borel_Sets by FINANCE1:8;
then [.k,k.] in Borel_Sets;
hence thesis by XXREAL_1:17;
end;
theorem
for k1,k2 being Real holds ].k1,k2.] is Event of Borel_Sets
proof
let k1,k2 be Real;
A1: [.k1,k2.] is Element of Borel_Sets by FINANCE1:8;
A2: {k1} is Event of Borel_Sets by Th1;
reconsider k1,k2 as ExtReal;
per cases;
suppose B1: k1<=k2;
[.k1,k2.] \ {k1} is Element of Borel_Sets by A1,A2,PROB_1:24;
hence thesis by B1,XXREAL_1:134;
end;
suppose k1>k2; then
].k1,k2.]={} by XXREAL_1:26;
hence thesis by PROB_1:4;
end;
end;
definition
func Family_of_halflines2 -> Subset-Family of REAL equals
the set of all right_closed_halfline(r) where r is Element of REAL;
coherence
proof
the set of all right_closed_halfline(r)
where r is Element of REAL c= bool REAL
proof
let p be object;
assume p in the set of all right_closed_halfline(r) where
r is Element of REAL;
then ex r being Element of REAL st p = right_closed_halfline(r);
hence p in bool REAL;
end;
hence thesis;
end;
end;
theorem
for Exy being ExtReal holds
{Exy} is Subset of ExtREAL by XXREAL_0:def 1,ZFMISC_1:31;
theorem
for Y being set, k being Nat st Y = REAL \ {k} holds
Y is Event of Borel_Sets
proof
let Y be set,k be Nat;
assume A1: Y = REAL \ {k};
reconsider Exx = k as Element of REAL by ORDINAL1:def 12,NUMBERS:19;
reconsider Z = {Exx} as Subset of REAL;
Z` is Element of Borel_Sets by PROB_1:15,Th1;
hence thesis by A1;
end;
reserve Exx for Real;
theorem
ex A being SetSequence of NAT st for n being Nat holds A.n = {n}
proof
deffunc U(Nat) = {In($1,NAT)};
AA: for x being Element of NAT holds U(x) in bool NAT
proof
let x be Element of NAT;
U(x) c= NAT by ZFMISC_1:31;
hence thesis;
end;
consider f being SetSequence of NAT such that
A1: for d be Element of NAT holds f.d = U(d) from FUNCT_2:sch 8(AA);
take f;
let n be Nat;
reconsider n as Element of NAT by ORDINAL1:def 12;
f.n = U(n) by A1;
hence thesis;
end;
theorem
for A being SetSequence of NAT st
(for n being Nat holds A.n = {n}) holds
for n being Nat holds (Partial_Union A).n in Borel_Sets
proof
let A be SetSequence of NAT;
assume A0: for n being Nat holds A.n={n};
defpred J[Nat] means (Partial_Union A).$1 in Borel_Sets;
S0: (Partial_Union A).0=A.0 by PROB_3:def 2;
(Partial_Union A).0 in Borel_Sets
proof
A.0={0} by A0;
hence thesis by Th1,S0;
end; then
J0: J[0];
J1: for n being Nat st J[n] holds J[n+1]
proof
let n be Nat;
assume Q0: J[n];
(Partial_Union A).(n+1) in Borel_Sets
proof
Q00: (Partial_Union A).(n+1) = (Partial_Union A).n \/ A.(n+1)
by PROB_3:def 2;
{n+1} in Borel_Sets by Th1; then
A.(n+1) in Borel_Sets by A0;
hence thesis by Q0,PROB_1:3,Q00;
end;
hence thesis;
end;
for n being Nat holds (Partial_Union A).n in Borel_Sets
proof
for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
hence thesis;
end;
hence thesis;
end;
theorem
REAL is Event of Borel_Sets
proof
reconsider A = {1} as Subset of REAL;
W1: A in Borel_Sets by Th1;
A` in Borel_Sets by Th1,PROB_1:15; then
A \/ A` in Borel_Sets by W1,PROB_1:3;
then [#]REAL in Borel_Sets by SUBSET_1:10;
hence thesis;
end;
Q00: ex A1 being SetSequence of REAL st for n being Nat holds A1.n={n}
proof
reconsider z = 1 as Element of REAL by NUMBERS:19;
deffunc U(Nat) = {z*$1};
consider f being SetSequence of REAL such that
A1:for d be Element of NAT holds f.d = U(d) from FUNCT_2:sch 4;
take f;
let n be Nat;
n in NAT by ORDINAL1:def 12; then
f.n = U(n) by A1 .= {n};
hence thesis;
end;
Q0: ex A being SetSequence of Borel_Sets st
for n being Nat holds A.n={n}
proof
consider A1 being SetSequence of REAL such that
A1: for n being Nat holds A1.n={n} by Q00;
A2: for n being Nat holds A1.n is Event of Borel_Sets
proof
let n be Nat;
reconsider n as Element of REAL by NUMBERS:19,ORDINAL1:def 12;
{n} in Borel_Sets by Th1;
hence thesis by A1;
end;
reconsider A1 as SetSequence of Borel_Sets by PROB_1:25,A2;
take A1;
thus thesis by A1;
end;
H2: ex A being SetSequence of REAL st for n being Nat holds A.n={-n}
proof
reconsider y = 1 as Element of REAL by NUMBERS:19;
consider z being Element of REAL such that A0: z=-y;
deffunc U(Nat) = {z*$1};
consider f being SetSequence of REAL such that
A1: for d be Element of NAT holds f.d = U(d) from FUNCT_2:sch 4;
take f;
let n be Nat;
n in NAT by ORDINAL1:def 12; then
f.n = U(n) by A1;
hence thesis by A0;
end;
theorem ZV5:
NAT is Event of Borel_Sets
proof
consider A being SetSequence of Borel_Sets such that
Q10: for n being Nat holds A.n={n} by Q0;
reconsider A as SetSequence of REAL;
Union A = NAT
proof
for x being object holds x in Union A iff x in NAT
proof
let x be object;
H0: for n being Element of NAT holds
for x being object holds x in {n} iff x=n by TARSKI:def 1;
H1: (ex n being Nat st x in A.n) implies
ex n being Element of NAT st x in A.n
proof
given n being Nat such that C0: x in A.n;
n+0 in NAT;
then consider n being Element of NAT such that C1: x in A.n by C0;
thus thesis by C1;
end;
thus x in Union A implies x in NAT
proof
assume C0: x in Union A;
ex n being Element of NAT st x in A.n & A.n={n}
proof
consider n being Element of NAT such that D0: x in A.n by C0,PROB_1:12,H1;
A.n = {n} by Q10;
hence thesis by D0;
end;
then ex n being Element of NAT st x=n by H0;
hence thesis;
end;
assume C000: x in NAT;
ex k being Nat st x in {k} & A.k={k}
proof
reconsider k = x as Nat by C000;
A: x in {k} by TARSKI:def 1;
A.k = {k} by Q10;
hence thesis by A;
end;
hence thesis by PROB_1:12;
end;
hence thesis;
end;
hence thesis by PROB_1:17;
end;
theorem
REAL \ NAT is Event of Borel_Sets
proof
consider Y being Subset of REAL such that A2: Y = NAT by NUMBERS:19;
Y` is Event of Borel_Sets by ZV5,A2,PROB_1:20;
hence thesis by A2;
end;
theorem ThA:
for AA being SetSequence of REAL holds
ex A being SetSequence of REAL st
for n being Nat holds A.n=(Partial_Union AA).n
proof
let AA be SetSequence of REAL;
deffunc U(Nat) = (Partial_Union AA).$1;
consider f being SetSequence of REAL such that
A1: for d be Element of NAT holds f.d = U(d) from FUNCT_2:sch 4;
take f;
let n be Nat;
n in NAT by ORDINAL1:def 12;
hence thesis by A1;
end;
theorem Th40:
INT is Event of Borel_Sets
proof
consider A1 being SetSequence of REAL such that
A1: for n being Nat holds A1.n={n} by Q00;
consider A2 being SetSequence of REAL such that
A2: for n being Nat holds A2.n={-n} by H2;
A3: for n being Nat holds (Partial_Union A1).n is Event of Borel_Sets
proof
defpred J[Nat] means (Partial_Union A1).$1 in Borel_Sets;
B0: (Partial_Union A1).0=A1.0 by PROB_3:def 2;
A1.0 = {0} by A1; then
J0: J[0] by Th1,B0;
J1: for n being Nat st J[n] holds J[n+1]
proof
let n be Nat;
assume Q0: J[n];
(Partial_Union A1).(n+1) in Borel_Sets
proof
C0: (Partial_Union A1).(n+1) = (Partial_Union A1).n \/ A1.(n+1)
by PROB_3:def 2;
C2: A1.(n+1) in Borel_Sets
proof
A1.(n+1)={n+1} by A1;
hence thesis by Th1;
end;
(Partial_Union A1).n \/ A1.(n+1) is Event of Borel_Sets
by Q0,C2,PROB_1:21;
hence thesis by C0;
end;
hence thesis;
end;
for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
hence thesis;
end;
defpred J[Nat] means (Partial_Union A2).$1 in Borel_Sets;
B0: (Partial_Union A2).0=A2.0 by PROB_3:def 2;
A2.0 = {-0} by A2; then
J0: J[0] by Th1,B0;
A4: for n being Nat holds
(Partial_Union A2).n is Event of Borel_Sets
proof
J1: for n being Nat st J[n] holds J[n+1]
proof
let n be Nat;
assume Q0: J[n];
(Partial_Union A2).(n+1) in Borel_Sets
proof
C0: (Partial_Union A2).(n+1) = (Partial_Union A2).n \/ A2.(n+1)
by PROB_3:def 2;
A2.(n+1)={-(n+1)} by A2; then
A2.(n+1) in Borel_Sets by Th1; then
(Partial_Union A2).n \/ A2.(n+1) is Event of Borel_Sets
by Q0,PROB_1:21;
hence thesis by C0;
end;
hence thesis;
end;
for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
hence thesis;
end;
consider B1 being SetSequence of REAL such that
A5: for n being Nat holds B1.n = (Partial_Union A1).n by ThA;
A6: for n being Nat holds B1.n is Event of Borel_Sets
proof
let n be Nat;
(Partial_Union A1).n is Event of Borel_Sets by A3;
hence thesis by A5;
end;
consider B2 being SetSequence of REAL such that
A7: for n being Nat holds B2.n = (Partial_Union A2).n by ThA;
A8: for n being Nat holds B2.n is Event of Borel_Sets
proof
let n be Nat;
(Partial_Union A2).n is Event of Borel_Sets by A4;
hence thesis by A7;
end;
reconsider B1 as SetSequence of Borel_Sets by A6,PROB_1:25;
reconsider B2 as SetSequence of Borel_Sets by A8,PROB_1:25;
A9: Union B1 is Event of Borel_Sets by PROB_1:26;
A10: Union B2 is Event of Borel_Sets by PROB_1:26;
Union B1 \/ Union B2 = INT
proof
for x being object holds x in (Union B1 \/ Union B2) iff x in INT
proof
let x be object;
thus x in Union B1 \/ Union B2 implies x in INT
proof
assume x in Union B1 \/ Union B2; then
per cases by XBOOLE_0:def 3;
suppose F0: x in Union B1;
consider n being Nat such that F1: x in B1.n by F0,PROB_1:12;
F2: x in (Partial_Union A1).n by F1,A5;
consider k being Nat such that
G1: k<=n & x in (Partial_Union A1).k by F2;
consider m being Nat such that
I0: m<=k & x in A1.m by G1,PROB_3:13;
x in INT
proof
x in {m} by I0,A1; then
I1: x=m by TARSKI:def 1;
m+0 in NAT;
hence thesis by I1,NUMBERS:17;
end;
hence thesis;
end;
suppose x in Union B2; then
consider k being Nat such that H1: x in B2.k by PROB_1:12;
x in (Partial_Union A2).k by H1,A7;
then consider z being Nat such that
G1: z<=k & x in A2.z by PROB_3:13;
x in {-z} by G1,A2;
then x=-z by TARSKI:def 1;
hence thesis by INT_1:def 1;
end;
end;
x in INT implies x in (Union B1 \/ Union B2)
proof
assume x in INT; then
consider k being Nat such that E0: x=k or x=-k by INT_1:def 1;
per cases by E0;
suppose x=k; then
reconsider p2 = x as Nat;
x in Union B1 \/ Union B2
proof
ex k being Nat st x in B1.k
proof
ex q being Nat st x in (Partial_Union A1).q
proof
x in {p2} by TARSKI:def 1; then
x in A1.p2 by A1; then
x in (Partial_Union A1).p2 by PROB_3:13;
hence thesis;
end; then
consider q being Nat such that I0: x in (Partial_Union A1).q;
B1.q=(Partial_Union A1).q by A5;
hence thesis by I0;
end;
then x in Union B1 by PROB_1:12;
hence thesis by XBOOLE_0:def 3;
end;
hence thesis;
end;
suppose F1: x=-k;
-k is Element of INT by INT_1:def 1;
then consider z being Element of INT such that G0: z=-k & -k=x by F1;
x in Union B1 \/ Union B2
proof
ex k being Nat st x in B2.k
proof
ex q being Nat st x in (Partial_Union A2).q
proof
K0: x in {-k} by G0,TARSKI:def 1;
x in A2.k by A2,K0; then
x in (Partial_Union A2).k by PROB_3:13;
then consider q being Nat such that I0: x in (Partial_Union A2).q;
thus thesis by I0;
end; then
consider q being Nat such that I0: x in (Partial_Union A2).q;
B2.q=(Partial_Union A2).q by A7;
hence thesis by I0;
end; then
x in Union B2 or x in Union B2 by PROB_1:12;
hence thesis by XBOOLE_0:def 3;
end;
hence thesis;
end;
end;
hence thesis;
end;
hence thesis;
end;
hence thesis by A9,A10,PROB_1:3;
end;
definition
let k be Nat;
let pm be Element of REAL;
func GoCross_Seq_REAL(pm,k) -> SetSequence of REAL means :Def4:
for n being Nat holds it.n = {pm*k*(n+1)"};
existence
proof
ex A being SetSequence of REAL st for n being Nat holds A.n={pm*k*(n+1)"}
proof
deffunc U(Element of NAT) = {pm*k*($1+1)"};
consider f being SetSequence of REAL such that
A1: for d be Element of NAT holds f.d = U(d) from FUNCT_2:sch 4;
take f;
let n be Nat;
reconsider n as Element of NAT by ORDINAL1:def 12;
f.n = U(n) by A1;
hence thesis;
end;
hence thesis;
end;
uniqueness
proof
let f1,f2 be SetSequence of REAL;
assume A1: for d being Nat holds f1.d={pm*k*(d+1)"};
assume A2: for d being Nat holds f2.d={pm*k*(d+1)"};
for d being Nat holds f1.d=f2.d
proof
let d be Nat;
f2.d={pm*k*(d+1)"} by A2;
hence thesis by A1;
end;
hence thesis;
end;
end;
definition
let k be Nat;
let pm be Element of REAL;
redefine func GoCross_Seq_REAL(pm,k) -> SetSequence of Borel_Sets;
correctness
proof
for n being Nat holds GoCross_Seq_REAL(pm,k).n is Event of Borel_Sets
proof
let n be Nat;
{pm*k*(n+1)"} in Borel_Sets by Th1;
hence thesis by Def4;
end;
hence thesis by PROB_1:25;
end;
end;
registration
let k be Nat;
let pm be Element of REAL;
cluster GoCross_Seq_REAL(pm,k) -> Borel_Sets-valued;
coherence;
end;
theorem
for pm being Element of REAL, k being Nat st
k>0 & pm<>0 holds GoCross_Seq_REAL(pm,k) is one-to-one
proof
let pm be Element of REAL;
let k be Nat;
assume A1: k>0 & pm<>0;
for x1,x2 being object st
x1 in dom GoCross_Seq_REAL(pm,k) &
x2 in dom GoCross_Seq_REAL(pm,k) &
GoCross_Seq_REAL(pm,k).x1 = GoCross_Seq_REAL(pm,k).x2 holds x1 = x2
proof
let x1,x2 be object;
assume B1: x1 in dom GoCross_Seq_REAL(pm,k);
assume B2: x2 in dom GoCross_Seq_REAL(pm,k);
assume B3: GoCross_Seq_REAL(pm,k).x1 = GoCross_Seq_REAL(pm,k).x2;
reconsider x1 as Nat by B1;
reconsider x2 as Nat by B2;
set d1 = pm*k*(x1+1)";
{pm*k*(x1+1)"}=GoCross_Seq_REAL(pm,k).x1 &
{pm*k*(x2+1)"}=GoCross_Seq_REAL(pm,k).x2 by Def4; then
B8: d1 in {pm*k*(x2+1)"} by TARSKI:def 1,B3;
(pm")*(pm*(k*(x1+1)"))=(pm")*(pm*(k*(x2+1)")) by TARSKI:def 1,B8; then
C1: (pm"*pm)*(k*(x1+1)")=(pm"*pm)*(k*(x2+1)");
pm"*pm=1 by A1,XCMPLX_0:def 7;
then k"*(k*(x1+1)")=k"*(k*(x2+1)") by C1; then
C2: k"*k*(x1+1)"=k"*k*(x2+1)";
k"*k=1 by A1,XCMPLX_0:def 7;
then x1+1 = x2+1 by C2,XCMPLX_1:201;
hence thesis;
end;
hence thesis by FUNCT_1:def 4;
end;
definition
let k be Nat;
let pm be Element of REAL;
func GoCross_Partial_Union(pm,k) -> SetSequence of REAL means :Def5:
it.0 = GoCross_Seq_REAL(pm,k).0 &
for n being Nat holds
it.(n+1) = it.n \/ GoCross_Seq_REAL(pm,k).(n+1);
existence
proof
defpred P[set,set,set] means
for x,y being Subset of REAL,s being Nat st
s = $1 & x = $2 & y = $3 holds y = x \/ GoCross_Seq_REAL(pm,k).(s+1);
A1: for n being Nat for x being Subset of REAL
ex y being Subset of REAL st P[n,x,y]
proof
let n be Nat;
let x be Subset of REAL;
take x \/ GoCross_Seq_REAL(pm,k).(n+1);
thus thesis;
end;
consider F being SetSequence of REAL such that
A2: F.0 = GoCross_Seq_REAL(pm,k).0 and
A3: for n being Nat holds P[n,F.n,F.(n+1)] from RECDEF_1:sch 2(A1);
take F;
thus F.0 = GoCross_Seq_REAL(pm,k).0 by A2;
let n be Nat;
thus thesis by A3;
end;
uniqueness
proof
let S1,S2 be SetSequence of REAL such that
A4: S1.0 = GoCross_Seq_REAL(pm,k).0 and
A5: for n being Nat holds S1.(n+1) = S1.n \/ GoCross_Seq_REAL(pm,k).(n+1)
and
A6: S2.0 = GoCross_Seq_REAL(pm,k).0 and
A7: for n being Nat holds S2.(n+1) = S2.n \/ GoCross_Seq_REAL(pm,k).(n+1);
defpred P[object] means S1.$1 = S2.$1;
for n being object holds n in NAT implies P[n]
proof
let n be object;
assume n in NAT;
then reconsider n as Element of NAT;
A8: for k2 being Nat st P[k2] holds P[k2+1]
proof
let k2 be Nat;
assume S1.k2 = S2.k2;
hence S1.(k2+1) = S2.k2 \/ GoCross_Seq_REAL(pm,k).(k2+1) by A5
.= S2.(k2+1) by A7;
end;
A9: P[0] by A4,A6;
for k being Nat holds P[k] from NAT_1:sch 2(A9,A8);
then S1.n = S2.n;
hence thesis;
end;
hence thesis;
end;
end;
definition
let k be Nat;
let pm be Element of REAL;
redefine func GoCross_Partial_Union(pm,k) -> SetSequence of Borel_Sets;
correctness
proof
defpred J[Nat] means
GoCross_Partial_Union(pm,k).$1 is Event of Borel_Sets;
GoCross_Seq_REAL(pm,k).0 is Event of Borel_Sets by PROB_1:25; then
J1: J[0] by Def5;
J2: for n being Nat st J[n] holds J[n+1]
proof
let n be Nat;
assume B1: J[n];
C2: GoCross_Seq_REAL(pm,k).(n+1) is Event of Borel_Sets by PROB_1:25;
GoCross_Partial_Union(pm,k).n \/ GoCross_Seq_REAL(pm,k).(n+1)
is Event of Borel_Sets by B1,C2,PROB_1:21;
hence thesis by Def5;
end;
for n being Nat holds J[n] from NAT_1:sch 2(J1,J2);
hence thesis by PROB_1:25;
end;
end;
registration
let k be Nat;
let pm be Element of REAL;
cluster GoCross_Partial_Union(pm,k) -> Borel_Sets-valued;
coherence;
end;
registration
let k be Nat;
let pm be Element of REAL;
cluster GoCross_Partial_Union(pm,k) -> non-descending;
coherence
proof
for s,q being Nat st s<=q holds
(GoCross_Partial_Union(pm,k)).s c= (GoCross_Partial_Union(pm,k)).q
proof
let s,q be Nat;
assume A1: s<=q;
consider j being Nat such that S2: q = s + j by A1,NAT_1:10;
defpred J[Nat] means
GoCross_Partial_Union(pm,k).s c= GoCross_Partial_Union(pm,k).(s+$1);
J1: J[0];
J2: for n being Nat st J[n] holds J[n+1]
proof
let n be Nat;
assume B1: J[n];
B20: (GoCross_Partial_Union(pm,k)).((s+n)+1) =
(GoCross_Partial_Union(pm,k)).((s+n)) \/
(GoCross_Seq_REAL(pm,k)).((s+n)+1) by Def5;
(GoCross_Partial_Union(pm,k)).s c=
(GoCross_Partial_Union(pm,k)).(s+(n+1)) by B1,XBOOLE_0:def 3,B20;
hence thesis;
end;
for n being Nat holds J[n] from NAT_1:sch 2(J1,J2);
hence thesis by S2;
end;
hence thesis;
end;
end;
definition
let pm be Element of REAL;
func GoCross_Union(pm) -> SetSequence of REAL means :Def6:
it.0 = Union GoCross_Partial_Union(pm,0) &
for n being Nat holds
it.(n+1) = it.n \/ Union GoCross_Partial_Union(pm,(n+1));
existence
proof
defpred P[set,set,set] means
for x,y being Subset of REAL,s being Nat st
s = $1 & x = $2 & y = $3 holds
y = x \/ Union GoCross_Partial_Union(pm,(s+1));
A1: for n being Nat for x being Subset of REAL
ex y being Subset of REAL st P[n,x,y]
proof
let n be Nat;
let x be Subset of REAL;
take x \/ Union GoCross_Partial_Union(pm,(n+1));
thus thesis;
end;
consider F being SetSequence of REAL such that
A2: F.0 = Union GoCross_Partial_Union(pm,0) and
A3: for n being Nat holds P[n,F.n,F.(n+1)] from RECDEF_1:sch 2(A1);
take F;
thus F.0 = Union GoCross_Partial_Union(pm,0) by A2;
let n be Nat;
thus thesis by A3;
end;
uniqueness
proof
let S1,S2 be SetSequence of REAL such that
A4: S1.0 = Union GoCross_Partial_Union(pm,0) and
A5: for n being Nat holds
S1.(n+1) = S1.n \/ Union GoCross_Partial_Union(pm,(n+1)) and
A6: S2.0 = Union GoCross_Partial_Union(pm,0) and
A7: for n being Nat holds
S2.(n+1) = S2.n \/ Union GoCross_Partial_Union(pm,(n+1));
defpred P[object] means S1.$1 = S2.$1;
for n being object holds n in NAT implies P[n]
proof
let n be object;
assume n in NAT;
then reconsider n as Nat;
A8: for k2 being Nat st P[k2] holds P[k2+1]
proof
let k2 be Nat;
assume S1.k2 = S2.k2;
hence S1.(k2+1) = S2.k2 \/ Union GoCross_Partial_Union(pm,(k2+1))
by A5
.= S2.(k2+1) by A7;
end;
A9: P[0] by A4,A6;
for k being Nat holds P[k] from NAT_1:sch 2(A9,A8);
then S1.n = S2.n;
hence thesis;
end;
hence thesis;
end;
end;
definition
let pm be Element of REAL;
redefine func GoCross_Union(pm) -> SetSequence of Borel_Sets;
correctness
proof
defpred J[Nat] means
(GoCross_Union(pm)).$1 is Event of Borel_Sets;
Union GoCross_Partial_Union(pm,0) is Event of Borel_Sets by PROB_1:17; then
J1: J[0] by Def6;
J2: for n being Nat st J[n] holds J[n+1]
proof
let n be Nat;
assume B1: J[n];
C2: Union GoCross_Partial_Union(pm,(n+1)) is Event of Borel_Sets by PROB_1:17;
(GoCross_Union(pm)).n \/ Union GoCross_Partial_Union(pm,(n+1))
is Event of Borel_Sets by B1,C2,PROB_1:21;
hence thesis by Def6;
end;
for n being Nat holds J[n] from NAT_1:sch 2(J1,J2);
hence thesis by PROB_1:25;
end;
end;
registration
let pm be Element of REAL;
cluster GoCross_Union(pm) -> Borel_Sets-valued;
coherence;
end;
registration
let pm be Element of REAL;
cluster GoCross_Union(pm) -> non-descending;
coherence
proof
for s,q being Nat st s<=q holds
(GoCross_Union(pm)).s c= (GoCross_Union(pm)).q
proof
let s,q be Nat;
assume s<=q; then
consider j being Nat such that S2: q = s + j by NAT_1:10;
defpred J[Nat] means
(GoCross_Union(pm)).s c= (GoCross_Union(pm)).(s+$1);
J1: J[0];
J2: for n being Nat st J[n] holds J[n+1]
proof
let n be Nat;
assume B1: J[n];
B20: (GoCross_Union(pm)).((s+n)+1) = (GoCross_Union(pm)).((s+n)) \/
Union GoCross_Partial_Union(pm,((s+n)+1)) by Def6;
(GoCross_Union(pm)).(s+n) c=
(GoCross_Union(pm)).(s+(n+1)) by XBOOLE_0:def 3, B20;
hence thesis by B1,XBOOLE_1:1;
end;
for n being Nat holds J[n] from NAT_1:sch 2(J1,J2);
hence thesis by S2;
end;
hence thesis;
end;
end;
theorem Th3:
for mym,myp being Element of REAL st mym=1 & myp=-1 holds
Union GoCross_Union(mym) \/ Union GoCross_Union(myp) = RAT
proof
let mym,myp be Element of REAL;
assume A: mym=1 & myp=-1;
for x being object holds
x in (Union GoCross_Union(mym) \/ Union GoCross_Union(myp)) iff x in RAT
proof
let x be object;
B1: x in (Union GoCross_Union(mym) \/
Union GoCross_Union(myp)) implies x in RAT
proof
assume x in (Union GoCross_Union(mym) \/ Union GoCross_Union(myp));
then per cases by XBOOLE_0:def 3;
suppose x in Union GoCross_Union(mym);
then consider k being Nat such that
D1: x in (GoCross_Union(mym)).k by PROB_1:12;
per cases;
suppose k=0;
then x in Union GoCross_Partial_Union(mym,0) by D1,Def6;
then consider k2 being Nat such that
E1: x in GoCross_Partial_Union(mym,0).k2 by PROB_1:12;
per cases;
suppose k2=0;
then x in GoCross_Seq_REAL(mym,0).0 by E1,Def5;
then x in {mym*0*(0+1)"} by Def4;
then x in INT by INT_1:def 1;
hence thesis by NUMBERS:14;
end;
suppose k2>0;
then consider n being Nat such that H1: k2=n+1 by NAT_1:6;
x in GoCross_Partial_Union(mym,0).n \/ GoCross_Seq_REAL(mym,0).(n+1)
by E1,H1,Def5; then
per cases by XBOOLE_0:def 3;
suppose H1: x in GoCross_Partial_Union(mym,0).n;
defpred J[Nat] means
x in GoCross_Partial_Union(mym,0).$1 implies x in RAT;
x in RAT
proof
J0: J[0]
proof
assume x in GoCross_Partial_Union(mym,0).0; then
x in GoCross_Seq_REAL(mym,0).0 by Def5; then
x in {mym*0*(0+1)"} by Def4; then
x=0 by TARSKI:def 1;
hence thesis by NUMBERS:18;
end;
J1: for n being Nat st J[n] holds J[n+1]
proof
let n be Nat;
assume K1: J[n];
assume x in GoCross_Partial_Union(mym,0).(n+1); then
x in (GoCross_Partial_Union(mym,0).n \/ GoCross_Seq_REAL(mym,0).(n+1))
by Def5; then
per cases by XBOOLE_0:def 3;
suppose x in GoCross_Partial_Union(mym,0).n;
hence thesis by K1;
end;
suppose x in GoCross_Seq_REAL(mym,0).(n+1);
then x in {mym*0*((n+1)+1)"} by Def4;
then x in INT by INT_1:def 1;
hence thesis by NUMBERS:14;
end;
end;
for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
hence thesis by H1;
end;
hence thesis;
end;
suppose x in GoCross_Seq_REAL(mym,0).(n+1);
then x in {mym*0*((n+1)+1)"} by Def4;
then x in INT by INT_1:def 1;
hence thesis by NUMBERS:14;
end;
end;
end;
suppose k>0;
then consider kh being Nat such that H1: k=kh+1 by NAT_1:6;
x in (GoCross_Union(mym)).kh
\/ Union GoCross_Partial_Union(mym,kh+1) by D1,H1,Def6; then
per cases by XBOOLE_0:def 3;
suppose J00: x in (GoCross_Union(mym)).kh;
defpred J[Nat] means x in (GoCross_Union(mym)).$1 implies x in RAT;
J0: J[0]
proof
assume x in (GoCross_Union(mym)).0;
then x in Union GoCross_Partial_Union(mym,0) by Def6;
then consider k2 being Nat such that
E1: x in GoCross_Partial_Union(mym,0).k2 by PROB_1:12;
per cases;
suppose k2=0;
then x in GoCross_Seq_REAL(mym,0).0 by E1,Def5;
then x in {mym*0*(0+1)"} by Def4;
then x in INT by INT_1:def 1;
hence thesis by NUMBERS:14;
end;
suppose k2>0;
then consider n being Nat such that H1: k2=n+1 by NAT_1:6;
x in GoCross_Partial_Union(mym,0).n \/ GoCross_Seq_REAL(mym,0).(n+1)
by E1,H1,Def5; then
per cases by XBOOLE_0:def 3;
suppose H1: x in GoCross_Partial_Union(mym,0).n;
defpred J[Nat] means
x in GoCross_Partial_Union(mym,0).$1 implies x in RAT;
J0: J[0]
proof
assume x in GoCross_Partial_Union(mym,0).0;
then x in GoCross_Seq_REAL(mym,0).0 by Def5;
then x in {mym*0*(0+1)"} by Def4;
then x=0 by TARSKI:def 1;
hence thesis by NUMBERS:18;
end;
J1: for n being Nat st J[n] holds J[n+1]
proof
let n be Nat;
assume K1: J[n];
assume x in GoCross_Partial_Union(mym,0).(n+1); then
x in (GoCross_Partial_Union(mym,0).n \/
GoCross_Seq_REAL(mym,0).(n+1)) by Def5; then
per cases by XBOOLE_0:def 3;
suppose x in GoCross_Partial_Union(mym,0).n;
hence thesis by K1;
end;
suppose x in GoCross_Seq_REAL(mym,0).(n+1);
then x in {mym*0*((n+1)+1)"} by Def4;
then x in INT by INT_1:def 1;
hence thesis by NUMBERS:14;
end;
end;
for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
hence thesis by H1;
end;
suppose x in GoCross_Seq_REAL(mym,0).(n+1);
then x in {mym*0*((n+1)+1)"} by Def4;
then x in INT by INT_1:def 1;
hence thesis by NUMBERS:14;
end;
end;
end;
J1:for n being Nat st J[n] holds J[n+1]
proof
let n be Nat;
assume K0000: J[n];
x in (GoCross_Union(mym)).(n+1) implies x in RAT
proof
assume x in (GoCross_Union(mym)).(n+1); then
x in (GoCross_Union(mym)).n \/
Union GoCross_Partial_Union(mym,(n+1)) by Def6; then
per cases by XBOOLE_0:def 3;
suppose x in (GoCross_Union(mym)).n;
hence thesis by K0000;
end;
suppose x in Union GoCross_Partial_Union(mym,(n+1));
then consider k being Nat such that
H1: x in GoCross_Partial_Union(mym,(n+1)).k by PROB_1:12;
defpred J[Nat] means
x in GoCross_Partial_Union(mym,(n+1)).$1 implies x in RAT;
J0: J[0]
proof
assume x in GoCross_Partial_Union(mym,(n+1)).0;
then x in GoCross_Seq_REAL(mym,(n+1)).0 by Def5;
then x in {mym*(n+1)*(0+1)"} by Def4;
then x=1*(n+1) by TARSKI:def 1,A;
hence thesis by NUMBERS:17,NUMBERS:14;
end;
J1: for k being Nat st J[k] holds J[k+1]
proof
let k be Nat;
assume K0: J[k];
set o = n+1, o2 = k+2, o3 = k+1;
x in GoCross_Partial_Union(mym,(n+1)).(k+1) implies x in RAT
proof
assume x in GoCross_Partial_Union(mym,(n+1)).(k+1); then
x in GoCross_Partial_Union(mym,(n+1)).k \/
GoCross_Seq_REAL(mym,(n+1)).(k+1) by Def5; then
per cases by XBOOLE_0:def 3;
suppose x in GoCross_Partial_Union(mym,(n+1)).k;
hence thesis by K0;
end;
suppose x in GoCross_Seq_REAL(mym,(n+1)).(k+1); then
x in {mym*(n+1)*(o3+1)"} by Def4; then
OZ: x=(n+1)*(k+2)" by TARSKI:def 1,A;
reconsider o2,o as Element of INT by NUMBERS:17;
o/o2 is Element of RAT by RAT_1:def 1;
hence thesis by OZ;
end;
end;
hence thesis;
end;
for k being Nat holds J[k] from NAT_1:sch 2(J0,J1);
hence thesis by H1;
end;
end;
hence thesis;
end;
for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
hence thesis by J00;
end;
suppose x in Union GoCross_Partial_Union(mym,kh+1);
then consider q being Nat such that
JK0: x in GoCross_Partial_Union(mym,kh+1).q by PROB_1:12;
defpred J[Nat] means
x in GoCross_Partial_Union(mym,(kh+1)).$1 implies x in RAT;
J0: J[0]
proof
assume x in GoCross_Partial_Union(mym,(kh+1)).0;
then x in GoCross_Seq_REAL(mym,(kh+1)).0 by Def5;
then x in {mym*(kh+1)*(0+1)"} by Def4;
then x=mym*(kh+1)*(1)" by TARSKI:def 1;
hence thesis by A,NUMBERS:17,NUMBERS:14;
end;
J1: for k being Nat st J[k] holds J[k+1]
proof
let k be Nat;
assume K0: J[k];
set o=kh+1, o2=k+2, o3=k+1;
x in GoCross_Partial_Union(mym,(kh+1)).(k+1) implies x in RAT
proof
assume x in GoCross_Partial_Union(mym,(kh+1)).(k+1); then
x in GoCross_Partial_Union(mym,(kh+1)).k \/
GoCross_Seq_REAL(mym,(kh+1)).(k+1) by Def5; then
per cases by XBOOLE_0:def 3;
suppose x in GoCross_Partial_Union(mym,(kh+1)).k;
hence thesis by K0;
end;
suppose x in GoCross_Seq_REAL(mym,(kh+1)).(k+1); then
x in {mym*(kh+1)*(o3+1)"} by Def4; then
OZ: x=(kh+1)*(k+2)" by TARSKI:def 1,A;
reconsider o2,o as Element of INT by NUMBERS:17;
::: reconsider o as Element of INT by NUMBERS:17;
o/o2 is Element of RAT by RAT_1:def 1;
hence thesis by OZ;
end;
end;
hence thesis;
end;
for k being Nat holds J[k] from NAT_1:sch 2(J0,J1);
hence thesis by JK0;
end;
end;
end;
suppose x in Union GoCross_Union(myp);
then consider k being Nat such that D1: x in (GoCross_Union(myp)).k
by PROB_1:12;
per cases;
suppose k=0;
then x in Union GoCross_Partial_Union(myp,0) by D1,Def6; then
consider k2 being Nat such that E1: x in GoCross_Partial_Union(myp,0).k2
by PROB_1:12;
per cases;
suppose k2=0;
then x in GoCross_Seq_REAL(myp,0).0 by E1,Def5;
then x in {myp*0*(0+1)"} by Def4;
then x in INT by INT_1:def 1;
hence thesis by NUMBERS:14;
end;
suppose k2>0;
then consider n being Nat such that H1: k2=n+1 by NAT_1:6;
x in GoCross_Partial_Union(myp,0).n \/ GoCross_Seq_REAL(myp,0).(n+1)
by E1,H1,Def5; then
per cases by XBOOLE_0:def 3;
suppose H1: x in GoCross_Partial_Union(myp,0).n;
defpred J[Nat] means
x in GoCross_Partial_Union(myp,0).$1 implies x in RAT;
J0: J[0]
proof
assume x in GoCross_Partial_Union(myp,0).0;
then x in GoCross_Seq_REAL(myp,0).0 by Def5;
then x in {myp*0*(0+1)"} by Def4;
then x=0 by TARSKI:def 1;
hence thesis by NUMBERS:18;
end;
x in RAT
proof
J1: for n being Nat st J[n] holds J[n+1]
proof
let n be Nat;
assume K1: J[n];
assume x in GoCross_Partial_Union(myp,0).(n+1); then
x in (GoCross_Partial_Union(myp,0).n \/ GoCross_Seq_REAL(myp,0).(n+1))
by Def5; then
per cases by XBOOLE_0:def 3;
suppose x in GoCross_Partial_Union(myp,0).n;
hence thesis by K1;
end;
suppose x in GoCross_Seq_REAL(myp,0).(n+1);
then x in {myp*0*((n+1)+1)"} by Def4;
then x in INT by INT_1:def 1;
hence thesis by NUMBERS:14;
end;
end;
for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
hence thesis by H1;
end;
hence thesis;
end;
suppose x in GoCross_Seq_REAL(myp,0).(n+1);
then x in {myp*0*((n+1)+1)"} by Def4;
then x in INT by INT_1:def 1;
hence thesis by NUMBERS:14;
end;
end;
end;
suppose k>0;
then consider kh being Nat such that H1: k=kh+1 by NAT_1:6;
x in (GoCross_Union(myp)).kh
\/ Union GoCross_Partial_Union(myp,kh+1) by D1,H1,Def6; then
per cases by XBOOLE_0:def 3;
suppose J00: x in (GoCross_Union(myp)).kh;
defpred J[Nat] means x in (GoCross_Union(myp)).$1 implies x in RAT;
J0: J[0]
proof
assume x in (GoCross_Union(myp)).0;
then x in Union GoCross_Partial_Union(myp,0) by Def6;
then consider k2 being Nat such that
E1: x in GoCross_Partial_Union(myp,0).k2 by PROB_1:12;
per cases;
suppose k2=0;
then x in GoCross_Seq_REAL(myp,0).0 by E1,Def5;
then x in {myp*0*(0+1)"} by Def4;
then x in INT by INT_1:def 1;
hence thesis by NUMBERS:14;
end;
suppose k2>0; then
consider n being Nat such that H1: k2=n+1 by NAT_1:6;
x in GoCross_Partial_Union(myp,0).n \/ GoCross_Seq_REAL(myp,0).(n+1)
by E1,H1,Def5; then
per cases by XBOOLE_0:def 3;
suppose H1: x in GoCross_Partial_Union(myp,0).n;
x in RAT
proof
defpred J[Nat] means
x in GoCross_Partial_Union(myp,0).$1 implies x in RAT;
J0: J[0]
proof
assume x in GoCross_Partial_Union(myp,0).0;
then x in GoCross_Seq_REAL(myp,0).0 by Def5;
then x in {myp*0*(0+1)"} by Def4;
then x=0 by TARSKI:def 1;
hence thesis by NUMBERS:18;
end;
J1: for n being Nat st J[n] holds J[n+1]
proof
let n be Nat;
assume K1: J[n];
assume x in GoCross_Partial_Union(myp,0).(n+1); then
x in (GoCross_Partial_Union(myp,0).n \/
GoCross_Seq_REAL(myp,0).(n+1)) by Def5; then
per cases by XBOOLE_0:def 3;
suppose x in GoCross_Partial_Union(myp,0).n;
hence thesis by K1;
end;
suppose x in GoCross_Seq_REAL(myp,0).(n+1);
then x in {myp*0*((n+1)+1)"} by Def4;
then x in INT by INT_1:def 1;
hence thesis by NUMBERS:14;
end;
end;
for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
hence thesis by H1;
end;
hence thesis;
end;
suppose x in GoCross_Seq_REAL(myp,0).(n+1);
then x in {myp*0*((n+1)+1)"} by Def4;
then x in INT by INT_1:def 1;
hence thesis by NUMBERS:14;
end;
end;
end;
J1:for n being Nat st J[n] holds J[n+1]
proof
let n be Nat;
assume K0000: J[n];
x in (GoCross_Union(myp)).(n+1) implies x in RAT
proof
assume x in (GoCross_Union(myp)).(n+1); then
x in (GoCross_Union(myp)).n \/
Union GoCross_Partial_Union(myp,(n+1)) by Def6; then
per cases by XBOOLE_0:def 3;
suppose x in (GoCross_Union(myp)).n;
hence thesis by K0000;
end;
suppose x in Union GoCross_Partial_Union(myp,(n+1));
then consider k being Nat such that
H1: x in GoCross_Partial_Union(myp,(n+1)).k by PROB_1:12;
defpred J[Nat] means
x in GoCross_Partial_Union(myp,(n+1)).$1 implies x in RAT;
J0: J[0]
proof
assume x in GoCross_Partial_Union(myp,(n+1)).0;
then x in GoCross_Seq_REAL(myp,(n+1)).0 by Def5;
then x in {myp*(n+1)*(0+1)"} by Def4; then
HH: x=-1*(n+1) by TARSKI:def 1, A;
-(n+1) is Element of INT by INT_1:def 1;
hence thesis by HH,NUMBERS:14;
end;
J1: for k being Nat st J[k] holds J[k+1]
proof
let k be Nat;
assume K0: J[k];
x in GoCross_Partial_Union(myp,(n+1)).(k+1) implies x in RAT
proof
assume x in GoCross_Partial_Union(myp,(n+1)).(k+1); then
x in GoCross_Partial_Union(myp,(n+1)).k \/
GoCross_Seq_REAL(myp,(n+1)).(k+1) by Def5; then
per cases by XBOOLE_0:def 3;
suppose x in GoCross_Partial_Union(myp,(n+1)).k;
hence thesis by K0;
end;
suppose x in GoCross_Seq_REAL(myp,(n+1)).(k+1); then
x in {myp*(n+1)*(k+1+1)"} by Def4;
hence thesis by A,RAT_1:def 2;
end;
end;
hence thesis;
end;
for k being Nat holds J[k] from NAT_1:sch 2(J0,J1);
hence thesis by H1;
end;
end;
hence thesis;
end;
for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
hence thesis by J00;
end;
suppose x in Union GoCross_Partial_Union(myp,kh+1);
then consider q being Nat such that
JK0: x in GoCross_Partial_Union(myp,kh+1).q by PROB_1:12;
defpred J[Nat] means
x in GoCross_Partial_Union(myp,(kh+1)).$1 implies x in RAT;
J0: J[0]
proof
assume x in GoCross_Partial_Union(myp,(kh+1)).0;
then x in GoCross_Seq_REAL(myp,(kh+1)).0 by Def5;
then x in {myp*(kh+1)*(0+1)"} by Def4; then
HH: x=-(kh+1) by TARSKI:def 1,A;
-(kh+1) in INT by INT_1:def 1;
hence thesis by HH,NUMBERS:14;
end;
J1: for k being Nat st J[k] holds J[k+1]
proof
let k be Nat;
assume K0: J[k];
set o3 = k+1;
x in GoCross_Partial_Union(myp,(kh+1)).(k+1) implies x in RAT
proof
assume x in GoCross_Partial_Union(myp,(kh+1)).(k+1); then
x in GoCross_Partial_Union(myp,(kh+1)).k \/
GoCross_Seq_REAL(myp,(kh+1)).(k+1) by Def5; then
per cases by XBOOLE_0:def 3;
suppose x in GoCross_Partial_Union(myp,(kh+1)).k;
hence thesis by K0;
end;
suppose x in GoCross_Seq_REAL(myp,(kh+1)).(k+1); then
x in {myp*(kh+1)*(o3+1)"} by Def4;
hence thesis by A,RAT_1:def 2;
end;
end;
hence thesis;
end;
for k being Nat holds J[k] from NAT_1:sch 2(J0,J1);
hence thesis by JK0;
end;
end;
end;
end;
x in RAT implies
x in (Union GoCross_Union(mym) \/ Union GoCross_Union(myp))
proof
assume x in RAT;
then reconsider x as Rational;
consider m being Integer, k being Nat such that C0: k<>0 & x=m/k
by RAT_1:8;
consider m2 being Nat such that C2: m=m2 or m=-m2 by INT_1:2;
per cases by C2;
suppose S1: m=m2;
consider q2 being Nat such that F2: q2=k-1 by C0;
G1: x in GoCross_Seq_REAL(mym,m2).q2
proof
GoCross_Seq_REAL(mym,m2).q2 = {1*m2*(q2+1)"} by Def4,A;
hence thesis by C0,S1,F2,TARSKI:def 1;
end;
G2: x in (GoCross_Partial_Union(mym,m2)).q2
proof
per cases;
suppose q2=0;
hence thesis by Def5,G1;
end;
suppose q2>0;
then consider q3 being Nat such that F2: q3=q2-1;
x in (GoCross_Partial_Union(mym,m2)).q2
proof
(GoCross_Partial_Union(mym,m2)).(q3+1)=
(GoCross_Partial_Union(mym,m2)).q3 \/
(GoCross_Seq_REAL(mym,m2)).(q3+1) by Def5;
hence thesis by F2,G1,XBOOLE_0:def 3;
end;
hence thesis;
end;
end;
x in (GoCross_Union(mym)).m2
proof
per cases;
suppose I1: m2=0;
x in Union GoCross_Partial_Union(mym,m2) by G2,PROB_1:12;
hence thesis by I1,Def6;
end;
suppose m2>0;
then consider m3 being Nat such that F2: m3=m2-1;
I2: x in(GoCross_Union(mym)).m3 or
x in Union GoCross_Partial_Union(mym,m3+1) by F2,G2,PROB_1:12;
x in (GoCross_Union(mym)).m2
proof
(GoCross_Union(mym)).(m3+1) = (GoCross_Union(mym)).m3 \/
Union (GoCross_Partial_Union(mym,m3+1)) by Def6;
hence thesis by I2,XBOOLE_0:def 3,F2;
end;
hence thesis;
end;
end;
then x in Union GoCross_Union(mym) by PROB_1:12;
hence thesis by XBOOLE_0:def 3;
end;
suppose S1: m=-m2;
consider q2 being Nat such that F2: q2=k-1 by C0;
G1: x in GoCross_Seq_REAL(myp,m2).q2
proof
GoCross_Seq_REAL(myp,m2).q2 = {(-1)*m2*(q2+1)"} by Def4,A;
hence thesis by C0,S1,F2,TARSKI:def 1;
end;
G2: x in (GoCross_Partial_Union(myp,m2)).q2
proof
per cases;
suppose q2=0;
hence thesis by Def5,G1;
end;
suppose q2>0;
then consider q3 being Nat such that F2: q3=q2-1;
x in (GoCross_Partial_Union(myp,m2)).q2
proof
(GoCross_Partial_Union(myp,m2)).(q3+1) =
(GoCross_Partial_Union(myp,m2)).q3 \/
(GoCross_Seq_REAL(myp,m2)).(q3+1) by Def5;
hence thesis by F2,G1,XBOOLE_0:def 3;
end;
hence thesis;
end;
end;
x in (GoCross_Union(myp)).m2
proof
per cases;
suppose m2=0; then
(GoCross_Union(myp)).m2 = Union GoCross_Partial_Union(myp,m2) by Def6;
hence thesis by G2,PROB_1:12;
end;
suppose m2>0;
then consider m3 being Nat such that F2: m3=m2-1;
I2: x in(GoCross_Union(myp)).m3 or
x in Union GoCross_Partial_Union(myp,m3+1) by F2,G2,PROB_1:12;
x in (GoCross_Union(myp)).m2
proof
(GoCross_Union(myp)).(m3+1) = (GoCross_Union(myp)).m3 \/
Union (GoCross_Partial_Union(myp,m3+1)) by Def6;
hence thesis by I2,XBOOLE_0:def 3,F2;
end;
hence thesis;
end;
end;
then x in (Union GoCross_Union(mym)) or
x in (Union GoCross_Union(myp)) by PROB_1:12;
hence thesis by XBOOLE_0:def 3;
end;
end;
hence thesis by B1;
end;
hence thesis;
end;
theorem Th41:
RAT is Event of Borel_Sets
proof
reconsider mym = 1 as Element of REAL by Ko1;
consider myp being Element of REAL such that A10: myp=-mym;
A0: Union GoCross_Union(mym) \/ Union GoCross_Union(myp) = RAT by A10,Th3;
A1: Union GoCross_Union(mym) is Event of Borel_Sets by PROB_1:17;
Union GoCross_Union(myp) is Event of Borel_Sets by PROB_1:17;
hence thesis by A1,PROB_1:21,A0;
end;
theorem
REAL \ INT is Event of Borel_Sets
proof
consider Y being Subset of REAL such that A2: Y = INT by NUMBERS:15;
Y` is Event of Borel_Sets by Th40,A2,PROB_1:20;
hence thesis by A2;
end;
theorem
REAL \ RAT is Event of Borel_Sets
proof
consider Y being Subset of REAL such that A2: Y = RAT by NUMBERS:12;
Y` is Event of Borel_Sets by Th41,A2,PROB_1:20;
hence thesis by A2;
end;
theorem
IRRAT is Event of Borel_Sets
proof
consider Y being Subset of REAL such that A2: Y = RAT by NUMBERS:12;
Y` is Event of Borel_Sets by Th41,A2,PROB_1:20;
hence thesis by A2;
end;
begin :: Construction of Borel-Sets
:: Since the Borel-Sets can be constructed by different sets of intervals,
:: we show, that other sets can do this.
theorem
Borel_Sets = sigma Family_of_halflines2
proof
A1: sigma Family_of_halflines2 c= sigma Family_of_halflines
proof
Family_of_halflines2 c= sigma(Family_of_halflines)
proof
for x being object holds x in Family_of_halflines2 implies
x in sigma Family_of_halflines
proof
let x be object;
assume x in Family_of_halflines2;
then consider r being Element of REAL such that
B1: x=right_closed_halfline(r);
x is Element of Borel_Sets by FINANCE1:3,B1;
hence thesis;
end;
hence thesis;
end;
hence thesis by PROB_1:def 9;
end;
sigma Family_of_halflines c= sigma Family_of_halflines2
proof
Family_of_halflines c= sigma Family_of_halflines2
proof
for x being object holds x in Family_of_halflines implies
x in sigma Family_of_halflines2
proof
let x be object;
assume x in Family_of_halflines;
then consider r being Element of REAL such that B1: x=halfline(r);
right_closed_halfline(r) is Event of sigma Family_of_halflines2
proof
set L = right_closed_halfline(r);
A3: L in Family_of_halflines2 & L=[.r,+infty.[;
Family_of_halflines2 c= sigma(Family_of_halflines2) by PROB_1:def 9;
hence thesis by A3;
end; then
(halfline(r))` is Event of sigma(Family_of_halflines2) by FINANCE1:2;
then ((halfline(r))`)` is Event of sigma(Family_of_halflines2)
by PROB_1:15;
hence thesis by B1;
end;
hence thesis;
end;
hence thesis by PROB_1:def 9;
end;
hence thesis by A1;
end;
begin :: Random-Variables for Stochastic Finance in Discrete Time
:: The aim is to show, that certain functions are Random-Variables, in order
:: to use theorems of Article RANDOM_3. An example for this application is the
:: definition of the arbitrage opportunity, which uses a Random-Variable
:: combined with a Probability-Measure.
reserve Omega,Omega2 for non empty set;
reserve Sigma for SigmaField of Omega;
reserve Sigma2 for SigmaField of Omega2;
reserve X,Y,Z for Function of Omega,REAL;
theorem
for X,Y being random_variable of Sigma,Borel_Sets holds
X+Y is random_variable of Sigma, Borel_Sets
proof
let X,Y be random_variable of Sigma, Borel_Sets;
reconsider X,Y as Real-Valued-Random-Variable of Sigma by RANDOM_3:9;
X+Y is Real-Valued-Random-Variable of Sigma;
hence thesis by RANDOM_3:9;
end;
theorem
for X,Y being random_variable of Sigma,Borel_Sets holds
X-Y is random_variable of Sigma, Borel_Sets
proof
let X,Y be random_variable of Sigma, Borel_Sets;
reconsider X,Y as Real-Valued-Random-Variable of Sigma by RANDOM_3:9;
X-Y is Real-Valued-Random-Variable of Sigma;
hence thesis by RANDOM_3:9;
end;
theorem
for X,Y being random_variable of Sigma,Borel_Sets holds
X(#)Y is random_variable of Sigma, Borel_Sets
proof
let X,Y be random_variable of Sigma, Borel_Sets;
reconsider X,Y as Real-Valued-Random-Variable of Sigma by RANDOM_3:9;
X(#)Y is Real-Valued-Random-Variable of Sigma;
hence thesis by RANDOM_3:9;
end;
theorem Th8:
for r being Real,
X being random_variable of Sigma,Borel_Sets holds
r(#)X is random_variable of Sigma, Borel_Sets
proof
let r be Real;
let Y be random_variable of Sigma, Borel_Sets;
reconsider Y as Real-Valued-Random-Variable of Sigma by RANDOM_3:9;
r(#)Y is Real-Valued-Random-Variable of Sigma;
hence thesis by RANDOM_3:9;
end;
theorem T:
for Omega,Omega2 be non empty set
for F be SigmaField of Omega
for F2 be SigmaField of Omega2
for k be Element of set_of_random_variables_on(F,F2) holds
Change_Element_to_Func(F,F2,k) is random_variable of F,F2
proof
let Omega,Omega2 be non empty set;
let F be SigmaField of Omega;
let F2 be SigmaField of Omega2;
let k be Element of set_of_random_variables_on(F,F2);
Change_Element_to_Func(F,F2,k) is Element of
{f where f is Function of Omega,Omega2:
f is (F,F2)-random_variable-like} by FINANCE1:def 7;
then Change_Element_to_Func(F,F2,k) in
{f where f is Function of Omega,Omega2:
f is (F,F2)-random_variable-like}; then
ex Y being Function of Omega,Omega2 st
Change_Element_to_Func(F,F2,k)=Y & Y is (F,F2)-random_variable-like;
hence thesis;
end;
theorem T1:
for Omega be non empty set
for F be SigmaField of Omega
for k be Element of set_of_random_variables_on(F,Borel_Sets) holds
ElementsOfPortfolioValueProb_fut(F,k) is
random_variable of F,Borel_Sets
proof
let Omega be non empty set;
let F be SigmaField of Omega;
let k be Element of set_of_random_variables_on(F,Borel_Sets);
ElementsOfPortfolioValueProb_fut(F,k) =
Change_Element_to_Func(F,Borel_Sets,k) by FINANCE1:def 8;
hence thesis by T;
end;
theorem
for p be Nat
for Omega, Omega2 be non empty set
for F be SigmaField of Omega
for F2 be SigmaField of Omega2
for G be sequence of set_of_random_variables_on(F,F2) holds
Element_Of(F,F2,G,p) is random_variable of F,F2
proof
let p be Nat;
let Omega, Omega2 be non empty set;
let F be SigmaField of Omega;
let F2 be SigmaField of Omega2;
let G be sequence of set_of_random_variables_on(F,F2);
G.p in set_of_random_variables_on(F,F2); then
consider Y being Function of Omega,Omega2 such that
A2: G.p=Y & Y is (F,F2)-random_variable-like;
Element_Of(F,F2,G,p)=Y by FINANCE1:def 9,A2;
hence thesis by A2;
end;
definition
let Omega be non empty set;
let F be SigmaField of Omega;
let X be non empty set;
let G be sequence of X;
let phi be Real_Sequence;
let n be Nat;
func RVElementsOfPortfolioValue_fut(phi,F,G,n) -> Function of Omega,REAL
means :Def5000:
for w being Element of Omega holds
it.w = ElementsOfPortfolioValueProb_fut(F,G.n).w * phi.n;
existence
proof
deffunc U(Element of Omega) =
In(ElementsOfPortfolioValueProb_fut(F,G.n).$1 * phi.n,REAL);
consider f being Function of Omega,REAL such that
A1: for d be Element of Omega holds f.d = U(d) from FUNCT_2:sch 4;
take f;
let n be Element of Omega;
f.n = U(n) by A1;
hence thesis;
end;
uniqueness
proof
let f1,f2 be Function of Omega,REAL;
assume that
A2: for w being Element of Omega holds f1.w =
ElementsOfPortfolioValueProb_fut(F,G.n).w * phi.n and
A3: for w being Element of Omega holds f2.w =
ElementsOfPortfolioValueProb_fut(F,G.n).w * phi.n;
for w being Element of Omega holds f1.w=f2.w
proof
let w be Element of Omega;
f2.w = ElementsOfPortfolioValueProb_fut(F,G.n).w * phi.n by A3;
hence thesis by A2;
end;
hence thesis;
end;
end;
theorem
for Omega be non empty set
for F be SigmaField of Omega
for G be sequence of set_of_random_variables_on(F,Borel_Sets)
for phi be Real_Sequence
for n be Nat holds
RVElementsOfPortfolioValue_fut(phi,F,G,n) is
random_variable of F,Borel_Sets
proof
let Omega be non empty set;
let F be SigmaField of Omega;
let G be sequence of set_of_random_variables_on(F,Borel_Sets);
let phi be Real_Sequence;
let n be Nat;
ElementsOfPortfolioValueProb_fut(F,G.n)
is random_variable of F,Borel_Sets by T1;
then
A1: phi.n(#)ElementsOfPortfolioValueProb_fut(F,G.n)
is random_variable of F,Borel_Sets by Th8;
for w being Element of Omega holds
(phi.n(#)ElementsOfPortfolioValueProb_fut(F,G.n)).w =
(RVElementsOfPortfolioValue_fut(phi,F,G,n)).w
proof
let w be Element of Omega;
phi.n * ElementsOfPortfolioValueProb_fut(F,G.n).w =
(phi.n(#)ElementsOfPortfolioValueProb_fut(F,G.n)).w by VALUED_1:6;
hence thesis by Def5000;
end;
hence thesis by A1,FUNCT_2:63;
end;
definition
let phi be Real_Sequence;
let Omega be non empty set;
let F be SigmaField of Omega;
let X be non empty set;
let G be sequence of X;
let w be Element of Omega;
func RVPortfolioValueFutExt_El(phi,F,G,w) -> Real_Sequence means
:Def5001:
for n being Nat holds
it.n = (RVElementsOfPortfolioValue_fut(phi,F,G,n)).w;
existence
proof
deffunc U(Element of NAT) =
(RVElementsOfPortfolioValue_fut(phi,F,G,$1)).w;
consider f being Real_Sequence such that
A1: for d be Element of NAT holds f.d = U(d) from FUNCT_2:sch 4;
take f;
let n be Nat;
reconsider n as Element of NAT by ORDINAL1:def 12;
f.n = U(n) by A1;
hence thesis;
end;
uniqueness
proof
let f1,f2 be Real_Sequence;
assume that
A1: for n being Nat holds
f1.n=(RVElementsOfPortfolioValue_fut(phi,F,G,n)).w and
A2: for n being Nat holds
f2.n=(RVElementsOfPortfolioValue_fut(phi,F,G,n)).w;
for n being Nat holds f1.n=f2.n
proof
let n be Nat;
f1.n=(RVElementsOfPortfolioValue_fut(phi,F,G,n)).w by A1;
hence thesis by A2;
end;
hence thesis;
end;
end;
definition
let d be Nat;
let phi be Real_Sequence;
let Omega be non empty set;
let F be SigmaField of Omega;
let X be non empty set;
let G be sequence of X;
let w be Element of Omega;
redefine func PortfolioValueFutExt(d,phi,F,G,w) -> Real equals
Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w))).d;
correctness
proof
defpred J[Nat] means
Partial_Sums(ElementsOfPortfolioValue_fut(phi,F,w,G)).$1=
Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w))).$1;
Partial_Sums(ElementsOfPortfolioValue_fut(phi,F,w,G)).0 =
ElementsOfPortfolioValue_fut(phi,F,w,G).0 by SERIES_1:def 1; then
Partial_Sums(ElementsOfPortfolioValue_fut(phi,F,w,G)).0 =
ElementsOfPortfolioValueProb_fut(F,G.0).w * phi.0 by FINANCE1:def 10; then
Partial_Sums(ElementsOfPortfolioValue_fut(phi,F,w,G)).0 =
(RVElementsOfPortfolioValue_fut(phi,F,G,0)).w by Def5000; then
Partial_Sums(ElementsOfPortfolioValue_fut(phi,F,w,G)).0 =
(RVPortfolioValueFutExt_El(phi,F,G,w)).0 by Def5001; then
J0: J[0] by SERIES_1:def 1;
J1: for n being Nat st J[n] holds J[n+1]
proof
let n be Nat;
assume B0: J[n];
B2: (ElementsOfPortfolioValue_fut(phi,F,w,G)).(n+1) =
(RVPortfolioValueFutExt_El(phi,F,G,w)).(n+1)
proof
(RVPortfolioValueFutExt_El(phi,F,G,w)).(n+1)=
(RVElementsOfPortfolioValue_fut(phi,F,G,n+1)).w by Def5001; then
(RVPortfolioValueFutExt_El(phi,F,G,w)).(n+1)=
ElementsOfPortfolioValueProb_fut(F,G.(n+1)).w * phi.(n+1) by Def5000;
hence thesis by FINANCE1:def 10;
end;
Partial_Sums(ElementsOfPortfolioValue_fut(phi,F,w,G)).(n+1)=
Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w))).n +
(ElementsOfPortfolioValue_fut(phi,F,w,G)).(n+1) by SERIES_1:def 1,B0;
hence thesis by B2,SERIES_1:def 1;
end;
for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
hence thesis;
end;
end;